How would the Bible advise a Christian to cope with fear?

I like the Psalm 23:4

Even though I walk
  through the darkest valley,
I will fear no evil,
  for you are with me;
your rod and your staff,
  they comfort me.

In Joshua 1:7-9, faith in God coupled with obedience is said to produce courage and overcome fear and discouragement.

“Be strong and very courageous. Be careful to obey all the law my servant Moses gave you; do not turn from it to the right or to the left, that you may be successful wherever you go. Keep this Book of the Law always on your lips; meditate on it day and night, so that you may be careful to do everything written in it. Then you will be prosperous and successful. Have I not commanded you? Be strong and courageous. Do not be afraid; do not be discouraged, for the Lord your God will be with you wherever you go.”

Proverbs 29:25 warns us to Fear God and not man:

Fear of man will prove to be a snare,
  but whoever trusts in the Lord is kept safe.

Psalm 56 relates an episode where David was afraid but trusted in God:

Be merciful to me, my God,
  for my enemies are in hot pursuit;
  all day long they press their attack.
My adversaries pursue me all day long;
  in their pride many are attacking me.

When I am afraid, I put my trust in you.
  In God, whose word I praise—
  in God I trust and am not afraid.
What can mere mortals do to me?

Indeed, if you open Psalms up at random and read a chapter, the odds are that the psalmist is dealing with fear.

Jesus advised people not to worry in Matthew 6:25-27:

“Therefore I tell you, do not worry about your life, what you will eat or drink; or about your body, what you will wear. Is not life more than food, and the body more than clothes? Look at the birds of the air; they do not sow or reap or store away in barns, and yet your heavenly Father feeds them. Are you not much more valuable than they? Can any one of you by worrying add a single hour to your life ?"

In Luke 12:4-7, Jesus tells us whom we should fear and why:

“I tell you, my friends, do not be afraid of those who kill the body and after that can do no more. But I will show you whom you should fear: Fear him who, after your body has been killed, has authority to throw you into hell. Yes, I tell you, fear him. Are not five sparrows sold for two pennies? Yet not one of them is forgotten by God. Indeed, the very hairs of your head are all numbered. Don’t be afraid; you are worth more than many sparrows.

It is also instructive to read about Gideon in Judges 6, who started out with very little courage, but because he obeyed the Lord, the Lord showed him things that increased his courage.

In 2 Kings 6:15-17, Elisha's faith permits him to trust that he is well defended by God, while his servant is terrified:

When the servant of the man of God got up and went out early the next morning, an army with horses and chariots had surrounded the city. “Oh no, my lord! What shall we do?” the servant asked.

“Don’t be afraid,” the prophet answered. “Those who are with us are more than those who are with them.”

And Elisha prayed, “Open his eyes, Lord, so that he may see.” Then the Lord opened the servant’s eyes, and he looked and saw the hills full of horses and chariots of fire all around Elisha.

Other stories that show fearful people learning courage from God are Moses in Exodus (at the burning bush) and Jacob in Genesis (when he is returning to his homeland and fears that his brother will kill him). But perhaps the most remarkable story is the entire Book of Acts, where a group of terrified fisherman and tax collectors who deserted their Lord stand up to Jerusalem and Rome to the point of death.

I must stop, because almost every Bible story that I can think of deals with courage: Mary agreeing to be the handmaid of the Lord, Joseph marrying her, and the Son of God consenting to be hung on a cross. Truly "The Fear of the Lord is the beginning of Wisdom."