What does the Bible say about vegetarian diets?

Rather than covering the same ground others have covered, I'll simply call your attention to Peter's vision in Acts 10:

9 About noon the following day as they were on their journey and approaching the city, Peter went up on the roof to pray. 10 He became hungry and wanted something to eat, and while the meal was being prepared, he fell into a trance. 11 He saw heaven opened and something like a large sheet being let down to earth by its four corners. 12 It contained all kinds of four-footed animals, as well as reptiles and birds. 13 Then a voice told him, “Get up, Peter. Kill and eat.” 14 “Surely not, Lord!” Peter replied. “I have never eaten anything impure or unclean.” 15 The voice spoke to him a second time, “Do not call anything impure that God has made clean.” 16 This happened three times, and immediately the sheet was taken back to heaven.

I don't believe in any of this stuff, but I'm willing to discuss the story. In Luke 24:41-43, the resurrected Jesus eats fish, to show that he's not a mere spirit:

... he said unto them, Have ye here any meat? And they gave him a piece of a broiled fish, and of an honeycomb. And he took it, and did eat before them.

This isn't a fish that he magically created, presumably never-alive. This is a regular dead cooked fish. Jesus is generally regarded as a positive example, so one might conclude that eating fish is ok.

Romans 14: 2-3 says

One believes he may eat anything, while the weak eats only vegetables. Let not him who eats despise him who abstains, and let not him who abstains pass judgement on him who eats; for God has welcomed him. (NIV)

which is clear enough to need no commentary.

In the beginning every animal and man were all vegetarian.

All Scripture is quoted from the King James translation, unless otherwise noted.

Gen 1:29 and 30 And God said, Behold, I have given you every herb bearing seed, which is upon the face of all the earth, and every tree, in the which is the fruit of a tree yielding seed; to you it shall be for meat. 30. And to every beast of the earth, and to every fowl of the air, and to every thing that creepeth upon the earth, wherein there is life, I have given every green herb for meat: and it was so.

After the Flood; God began carnivorous animals including man.

Genesis 9:2 through 6 And the fear of you and the dread of you shall be upon every beast of the earth, and upon every fowl of the air, upon all that moveth upon the earth, and upon all the fishes of the sea; into your hand are they delivered. 3.Every moving thing that liveth shall be meat for you; even as the green herb have I given you all things. 4.But flesh with the life thereof, which is the blood thereof, shall ye not eat. 5.And surely your blood of your lives will I require; at the hand of every beast will I require it, and at the hand of man; at the hand of every man's brother will I require the life of man. 6.Whoso sheddeth man's blood, by man shall his blood be shed: for in the image of God made he man.

After the Flood God set man and beasts at odds, so that they would always kill one another. We are not told why God decided to set man and the animals at odds, or even why he set man against man, but we may have some clues as to why he did this.

Genesis 6:5 through 7 And GOD saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart was only evil continually. 6.And it repented the LORD that he had made man on the earth, and it grieved him at his heart. 7.And the LORD said, I will destroy man whom I have created from the face of the earth; both man, and beast, and the creeping thing, and the fowls of the air; for it repenteth me that I have made them.

So God decided to destroy the Earth and all of its inhabitants.

Gen 6:11 through 13 The earth also was corrupt before God, and the earth was filled with violence. 12.And God looked upon the earth, and, behold, it was corrupt; for all flesh had corrupted his way upon the earth. 13.And God said unto Noah, The end of all flesh is come before me; for the earth is filled with violence through them; and, behold, I will destroy them with the earth.

But God decided to save Noah and enough animals to replenish the Earth.

Genesis 6:8 But Noah found grace in the eyes of the LORD.

Genesis 6:17 through 20 And, behold, I, even I, do bring a flood of waters upon the earth, to destroy all flesh, wherein is the breath of life, from under heaven; and every thing that is in the earth shall die. 18.But with thee will I establish my covenant; and thou shalt come into the ark, thou, and thy sons, and thy wife, and thy sons' wives with thee. 19.And of every living thing of all flesh, two of every sort shalt thou bring into the ark, to keep them alive with thee; they shall be male and female. 20.Of fowls after their kind, and of cattle after their kind, of every creeping thing of the earth after his kind, two of every sort shall come unto thee, to keep them alive.

And so it may be that due to man's ruination of the Earth that God decided to make the world a dangerous place for all of his creation, and through wariness of one another man could never again band together to ruin the Earth.

hope this helps

The bible portrays that God's original perfect design for his creation included that for humans be vegetarian:

God said, “Now we will make humans, and they will be like us. We will let them rule the fish, the birds, and all other living creatures.” - Genesis 1:26 CEV

'Rule' - neither constitutes killing nor eating, but rather governing and nurturing - or to use a biblical word, being good 'stewards' of the creation. More explicitly:

"I [God] have provided all kinds of fruit and grain for you [humans] to eat. [and that's it! No animal meat is mentioned.] - Genesis 1:29 CEV (bold emphasis and bracketed interpolations mine, cf. also Gen 2:16)

A consequence of mankind's rebellion against God was a fundamental disruption of the natural order - no longer would the fruits of the soil yield up their nutrition so easily:

17 The Lord said to the man, “You listened to your wife and ate fruit from that tree. And so, the ground will be under a curse because of what you did. As long as you live, you will have to struggle to grow enough food. 18 Your food will be plants, but the ground will produce thorns and thistles. 19 You will have to sweat to earn a living; you were made out of soil, and you will once again turn into soil.” - Genesis 3:17-19 CEV

With an ongoing escalation of the wickedness of mankind, the creation was increasingly subjected to further frustrations of its original purpose (cf. Romans 8:20) which included amongst other things (most) people abandoning vegetarianism, with God's permission to do so. (cf. Genesis 9:3) However though God gave this concession (perhaps due to the hardness of men's hearts as he would later give a concession for divorce), some of the greatest exemplars of faithfulness towards God in the Old Testament - Daniel and his colleagues - demonstrated that not only was it possible to subsist on a vegetarian diet, but that one could thrive under God's blessing in doing so. (cf. Daniel 1)

The 'good news' part of the bible, is that God did not leave mankind and His wider creation to continue the headlong rush in to sin, futility, death and decay; but rather sent His only begotten son that not only through him, would people be forgiven their sin and restored to relationship with their Creator (cf. John 3:16-17), but that the creation itself would be completely delivered from its oppression (cf. Revelation 21:1-5). This remains a work in progress, but we can see a glimpse of God's intended state for his redeemed creation in the following prophecy of Isaiah:

Peace at Last

1 Like a branch that sprouts from a stump,

someone from David’s family will someday be king.

2 The Spirit of the Lord will be with him

to give him understanding, wisdom, and insight.

He will be powerful, and he will know and honor the Lord.

3 His greatest joy will be to obey the Lord.

This king won’t judge by appearances or listen to rumors.

4 The poor and the needy will be treated with fairness and with justice.

His word will be law everywhere in the land,

and criminals will be put to death.

5 Honesty and fairness will be his royal robes.

6 Leopards will lie down with young goats,

and wolves will rest with lambs.

Calves and lions will eat together

and be cared for by little children.

7 Cows and bears will share the same pasture;

their young will rest side by side.

Lions and oxen will both eat straw.

8 Little children will play near snake holes.

They will stick their hands into dens of poisonous snakes and never be hurt.

9 Nothing harmful will take place on the Lord’s holy mountain.

Just as water fills the sea, the land will be filled

with people who know and honor the Lord. - Isaiah 11:1-9 CEV

i.e. Carnivorism will cease when God's purpose through the promised Messiah is completely fulfilled. In the mean time, the permissive stance towards meat-eating remains (God's priority is first to reconcile sinners to himself), but for those who have their own conviction for vegetarianism, the bible explicitly endorses that it is right for you to abide in your convictions (cf. Romans 14).

Since you did not state which denomination you'd like to find out from, I will provide you an answer from a purely sola scriptura point of view. All verse emphasis mine.

Taken wholly, the NKJV Bible does not contain the word vegetarian. However, there are a ton of verses about eating, the virtue of eating, what to eat and what not to eat. Let me give you some background in Genesis as it is the first few mentions of eating.

In the beginning, man and beasts were all vegans!

Genesis 1 New King James Version (NKJV)

The History of Creation

29 And God said, “See, I have given you every herb that yields seed which is on the face of all the earth, and every tree whose fruit yields seed; to you it shall be for food. 30 Also, to every beast of the earth, to every bird of the air, and to everything that creeps on the earth, in which there is life, I have given every green herb for food”; and it was so.

And so it was also in the Garden of Eden.

Genesis 2 New King James Version (NKJV)

Life in God’s Garden

8 The Lord God planted a garden eastward in Eden, and there He put the man whom He had formed. 9 And out of the ground the Lord God made every tree grow that is pleasant to the sight and good for food.

15 Then the Lord God took the man and put him in the garden of Eden to tend and keep it. 16 And the Lord God commanded the man, saying, “Of every tree of the garden you may freely eat;

However, after Noah's flood, things changed.

Genesis 9 New King James Version (NKJV)

God’s Promise to Noah

Every moving thing that lives shall be food for you. I have given you all things, even as the green herbs. 4 But you shall not eat flesh with its life, that is, its blood.

Verse 4 states very clearly that none of us should eat animals that are still alive. Thus, the animals have to be killed. This is the only reason that can justify why people kill animals for food. Also, the fact that animals have to be killed for our sustenance alludes to the fact that our relationship with God is tarnished. This can be seen as in Genesis, man did not need meat. You may read this link for a detailed explanation on the need to kill animals after the fall of Adam and Eve.

Now, since that Christians are able to eat meat and vegetables now, how and what should or should we not eat? Romans 14 in the New Testament explains it very well.

You can eat according to your own freewill, as long as you give thanks to God as He made it possible for you to be where you are, eating what you eat. If your heart is convinced that you should be a vegetarian, than go ahead. The Bible does not say that you can eat only meat. However, do not judge others that do not follow your dietary rules as;

Matthew 7 New King James Version (NKJV)

Do Not Judge

7 “Judge not, that you be not judged. 2 For with what judgment you judge, you will be judged; and with the measure you use, it will be measured back to you.

I will end this answer with the words of Jesus Christ in Mark 7.

17 When He had entered a house away from the crowd, His disciples asked Him concerning the parable. 18 So He said to them, “Are you thus without understanding also? Do you not perceive that whatever enters a man from outside cannot defile him, 19 because it does not enter his heart but his stomach, and is eliminated, thus purifying all foods?”[g] 20 And He said, “What comes out of a man, that defiles a man. 21 For from within, out of the heart of men, proceed evil thoughts, adulteries, fornications, murders, 22 thefts, covetousness, wickedness, deceit, lewdness, an evil eye, blasphemy, pride, foolishness. 23 All these evil things come from within and defile a man.”

There are a number of verses that talk about what you should or can eat and a few of them do concern vegetarian eating. There are a number of Christian sects that even have a "health message," where they prescribe a vegetarian diet. I think the most notable one is Seventh Day Adventists, but note that the diet is a suggestion in their mind, not Biblical law.

Starting from the beginning, there is a development of what God gives mankind to eat. In the garden, God first gives the fruit of the trees and any seed bearing plant (fruits).

Then God said, “I give you every seed-bearing plant on the face of the whole earth and every tree that has fruit with seed in it. They will be yours for food. And to all the beasts of the earth and all the birds in the sky and all the creatures that move along the ground—everything that has the breath of life in it—I give every green plant for food.” And it was so.
Genesis 1:29-30 (NIV)

When Creation was completed, God gave Adam any fruit he wanted as his food, and to all the other creatures, He gave any green plant. In Genesis 2:16-17 God gives Adam every tree in the Garden and forbids that Adam should eat from the Tree of Knowledge, but that is a theological/obedience issue rather than a practical, dietary issue.

During God's pronouncements of punishments for eating the forbidden fruit, Adam is told that his food will be plants in Genesis 3:18. It's a bit ambiguous, but we can assume that it means all plants, like the animals were first given.

After the fall we see an interesting development. The people became farmers and kept flocks.

Now Abel kept flocks, and Cain worked the soil.
Genesis 4:2 (NIV)

It also seems to have been permitted as well, that they should keep flocks, and, we assume, eat their meat, but that may be too hasty.

After Noah and his family survive the Flood, we see God allow animal flesh eating with only two prohibitions:

Everything that lives and moves about will be food for you. Just as I gave you the green plants, I now give you everything. But you must not eat meat that has its lifeblood still in it.
Genesis 9:3-4 (NIV)

Now God has given everything to eat, except blood and other people (that is in following verses and I left it out because you don't usually need to convince people that cannibalism is wrong).

This leaves a few questions about the previous verses. The first is did they (the righteous) actually eat meat before the Flood? We know they tended flocks, but they also sacrificed. It is possible that they only tended flocks for the sacrifice. The second question is when exactly did God give them the green plants? The earliest hint at this is after the fall in Genesis 3:18 and Genesis 4:2. God says plants will be your food, then Cain worked the soil later, which we assume means farming fruit and vegetables.

Some sects use the following verses and logic to support that vegetarian eating is better for you. Though that may be true, I think you would be hard pressed to say that the Bible commands a vegetarian diet, and I have never heard of a group that says it does, though, I do know that the SDA say it is encouraged.

There is another interesting story concerning food and vegetarian diets. Daniel 1 details a story where Daniel, a young man, was taken captive because he was intelligent, young, and handsome, and was given food from the royal table. Daniel saw this as an abomination, likely because the meat of that table included animals sacrificed to idols and might also have included blood. So Daniel asks his overlord to give him and the other servants only vegetables and water for 10 days and see if they are not healthier than the others.

At the end of the ten days they looked healthier and better nourished than any of the young men who ate the royal food.
Daniel 1:15 (NIV)

Some groups, like the SDA, point to this as factual and Biblical evidence that a vegetarian diet is better for you. Others say that it was the power of God that made them healthier because they wanted to avoid the royal table because it was defiled, not simply because it had meat on it; the objection was against meat that either had blood in it or was sacrificed to idol or both. They then develop ideas like a "Daniel fast," where you eat only vegetables and water for 10 days in an effort to increase your faith and the also tend to quote Mark 7:18-19, stating that no food can defile you. It's a bit of a red herring because Christians with the health message would agree.

Whatever any Christian decides, there is one verse that cannot be more clear on this subject:

So whether you eat or drink or whatever you do, do it all for the glory of God.
1 Corinthians 10:31 (NIV)