Why must we obey the ten commandments if Jesus died for our sins?

And one of them, versed in the Law, tested him by asking: 36 “Teacher, which is the greatest commandment in the Law?” 37 He said to him: “‘You must love Jehovah your God with your whole heart and with your whole soul and with your whole mind.’ 38 This is the greatest and first commandment. 39 The second, like it, is this: ‘You must love your neighbor as yourself.’ 40 On these two commandments the whole Law hangs, and the Prophets.” (Matthew 22:35-40 NWT)

Jesus said that the greatest commandments were to love God and neighbor. If you love God, you will not do the things that he hates like murder and stealing. If you love your neighbors you will not steal from them or hurt them. The law of Christ covers every commandment in the OT and beyond. As we study his word the Bible and learn more about him, we learn the principles found in his word and we are able to follow them. Jesus ransom covers the sacrificial parts and us having love covers the rest so that we don't need a list of do nots because we know what God hates and what would be unloving.

Go on carrying the burdens of one another, and in this way you will fulfill the law of the Christ. (Galatians 6:2 NWT)

The original commandment was to not eat of the fruit of the knowledge of function and dysfunction. When a person does eat of this fruit the result is pride and anxiety. The result of pride and anxiety is covetousness, blasphemy, murder, dishonor, alcohol abuse, and much many more things. The Ten Commandments enforce that the result of this pride and anxiety should not be done. Yet the real correction to Pride and Anxiety is Humility and Faith. Therefore, one who has Faith in God and is Humble, does not gain the result of pride and anxiety because they did not become prideful or anxiousness.

Logically then if one is not prideful or anxiousness then the result of being prideful or anxiousness does not occur, so the fulfillment of the law happens by nature. That does not mean that the commandments should be broken. It just means that they wont be broken.

Hebrews 11:6 tells us how to please God, and it's not through the Ten Commandments. "But without faith it is impossible to please Him, for he who comes to God must believe that He is, and that He is a rewarder of those who diligently seek Him."

None of us can please God through the Law, however: "Now we know that whatever the law says, it says to those who are under the law, that every mouth may be stopped, and all the world may become guilty before God" (Romans 3:19). In other words, the law shows us that we are sinners and we need an alternate way to please God, and that's when faith enters in. Through faith all of us may be saved (Romans 4:16), because all of us have been given a measure of faith.

So to please God, read the Bible, believe that through it Jesus speaks to you, and obey what it says to the best of your understanding.

Disclaimer: This answer is from a Wesleyan soteriological perspective.

Upon reading previous answers to this question, you could be forgiven for thinking that keeping the ten commandments is no longer necessary. Such a view would be entirely false. As has been already expressed in different ways, the law was given that we might know our sinful condition before God and turn to Christ for our salvation. The promise of salvation however, includes us receiving a new heart that is inclined towards fulfilling God's perfect law:

I will give them an undivided heart and put a new spirit in them; I will remove from them their heart of stone and give them a heart of flesh. Then they will follow my decrees and be careful to keep my laws. They will be my people, and I will be their God. (Ezekiel 11:19,20 NIV)

As well as God empowering believers to do this, the bible makes it clear that they must co-operate rather than resist the working of the Holy Spirit in this regard. Thus, Jesus instructs His disciples:

If you love me, keep my commands. (John 14:15 NIV)

Not everyone who says to Me, ‘Lord, Lord,’ shall enter the kingdom of heaven, but he who does the will of My Father in heaven. Many will say to Me in that day, ‘Lord, Lord, have we not prophesied in Your name, cast out demons in Your name, and done many wonders in Your name?’ And then I will declare to them, ‘I never knew you; depart from Me, you who practice lawlessness!’ (Matthew 7:21-23 NKJV)

And Stephen identified the problem with His persecutors as:

You stiff-necked people, uncircumcised in heart and ears, you always resist the Holy Spirit. As your fathers did, so do you (Acts 7:51 ESV - emphasis added)

It's not that keeping the commandments are a pre-condition for us receiving the free gift of salvation through faith in Christ, but keeping them are a necessary sign that we have in fact received this gift - otherwise, our 'faith' is proven false, self-deceptive and worthless. The Scriptures and the Holy Spirit give no assurance of salvation to those 'who practice lawlessness':

But be doers of the word, and not hearers only, deceiving yourselves. For if anyone is a hearer of the word and not a doer, he is like a man who looks intently at his natural face in a mirror. For he looks at himself and goes away and at once forgets what he was like. But the one who looks into the perfect law, the law of liberty, and perseveres, being no hearer who forgets but a doer who acts, he will be blessed in his doing. (James 1:22-25 ESV)

The 'perfect law' is summed up in the great commandment and its corollary:

“Teacher, which is the great commandment in the Law?” And he said to him, “You shall love the Lord your God with all your heart and with all your soul and with all your mind. This is the great and first commandment. And a second is like it: You shall love your neighbor as yourself. On these two commandments depend all the Law and the Prophets.” (Matthew 22:36-40 ESV)

It is greater than (a higher standard), but necessarily includes the ten commandments. In summary:

And by this we know that we have come to know him, if we keep his commandments. Whoever says “I know him” but does not keep his commandments is a liar, and the truth is not in him, but whoever keeps his word, in him truly the love of God is perfected. By this we may know that we are in him: whoever says he abides in him ought to walk in the same way in which he walked. (1 John 2:3-6 ESV)

I think your answer lies in The Threefold Uses of the Law, and more specifically, Romans 7:1,11,24

1 Yet if it had not been for the law, I would not have known sin. I would not have known what it is to covet if the law had not said, "You shall not covet." ... 11 For sin, seizing an opportunity through the commandment, deceived me and through it killed me. ... 24 Wretched man that I am! Who will deliver me from this body of death? Thanks be to God through Jesus Christ our Lord!

In brief, the Law was meant to show us our sin. While it is required that you obey The Law, you are, by your nature, unable to (Romans 8:7). This is the hopeless estate of man. Jesus saves us by accomplishing two things:

1. He perfectly obeys the Law, providing obedience that we are missing and
2. He suffers the penalty for our sin.

That is termed Christ's active and passive obedience, active obedience being obeying the whole moral law of God (not only the Ten Commandments) and passive obedience being his obedient suffering on the cross.

The natural reaction, as was the reaction of Erasmus to Luther, is, if we are saved by Christ's work, and need not obey the Ten Commandments, then do we not then give license to evil behaviour? This is the argument in Romans 6:1, and the answer is, "By no means! How can we who died to sin still live in it?"

What Paul says in Romans 6:2 is that if we are truly changed by Holy Spirit, we have spiritually died with Christ and been raised to new life in Christ, then our nature (that nature which causes us to sin) will remain in our flesh but replaced in our spirit, and so we are regenerate and changed, still struggling with sin, but hating sin. We see this in Romans 7:25.

May I say, you've touched on the heart of the Gospel, but provoked some of the deeper theological nuances surrounding it. May I encourage you to find someone you can sit with in a coffee shop and do a study of Romans? Maybe pick up Martin Luther's commentary on Romans?