I know that sin separates us from God and that God hates sin. Does God judge all sin the same (example: murder vs lying)?

Something we need to remember is that ALL sin separates us from God. In that sense, all sin is equal because, apart from reconciliation to God through Jesus Christ, all sin arrives at the same outcome. As Paul so famously wrote, For the wages of sin is death—Romans 6:23a. Those seven words apply to all sin, whether in thought, word or deed. Sin has impacted us all the same; as Paul also wrote, [S]in entered the world through one man, and death through sin, and in this way death came to all men, because all sinned—Paul, Romans 5:12. Death became a certainty because sin is a certainty. And to make sin even more pervasive, the apostle James noted, For whoever keeps the whole law and yet stumbles at just one point is guilty of breaking all of it—James 2:10. Those verses treat sin as a whole, underscoring what we said at the beginning: All sin separates us from God. In that context, therefore, all sin is equal. However, because of Jesus Christ, there is gracious remedy; as the apostle John wrote, If we confess our sins, he is faithful and just and will forgive us our sins and purify us from all unrighteousness—I John 1:9.
At the same time, it seems obvious that some sins are worse than others in both motivation and effects, and should be judged accordingly. Stealing a loaf of bread is vastly different than exterminating a million people. The Old Testament seems to reflect that understanding, as God applied different penalties to different sins, suggesting variations in the seriousness of some sins. A thief paid restitution; an occult practitioner was cut off from Israel; one who committed adultery or a homosexual act or cursed his parents was put to death (see Exodus 22 and Leviticus 20). In the New Testament Jesus said it would be easier on the day of judgment for Sodom than for Capernaum, a city along the shore of the Sea of Galilee and thought to be Jesus’ initial headquarters during the first part of His ministry. Why? Capernaum remained in unbelief and refusal to repent, even after witnessing His miracles. As Jesus Himself noted, And you, Capernaum, will you be lifted up to the skies? No, you will go down to the depths. If the miracles that were performed in you had been performed in Sodom, it would have remained to this day. But I tell you that it will be more bearable for Sodom on the day of judgment than for you”—Jesus Christ, Matthew 11:23-24. Incidentally, the Old Testament Ezekiel listed the sins for which Sodom was judged: Now this was the sin of your sister Sodom: She and her daughters were arrogant, overfed and unconcerned; they did not help the poor and needy. They were haughty and did detestable things before me. Therefore I did away with them as you have seen—God speaking, Ezekiel 16:49-50.
When Jesus spoke of his second coming and judgment, he warned that, among those deserving punishment, That servant who knows his master’s will and does not get ready or does not do what his master wants will be beaten with many blows. But the one who does not know and does things deserving punishment will be beaten with few blows. From everyone who has been given much, much will be demanded; and from the one who has been entrusted with much, much more will be asked—Jesus Christ, Luke 12:47-48. And Jesus was perhaps at His angriest when He was dealing with the pride and unbelief of the religious leaders.
In another, somewhat contrasting, view of “degree” of sin, in the Sermon on the Mount, Jesus makes this comparison: You have heard that it was said to the people long ago, ‘DO NOT MURDER, and anyone who murders will be subject to judgment.’ But I tell you that anyone who is angry with his brother will be subject to judgment. Again, anyone who says to his brother, ‘Raca,’ is answerable to the Sanhedrin. But anyone who says, ‘You fool!’ will be in danger of the fire of hell. . . . You have heard that it was said, ‘DO NOT COMMIT ADULTERY.’ But I tell you that anyone who looks at a woman lustfully has already committed adultery with her in his heart—Jesus Christ, the Sermon on the Mount, Matthew 5:21-22, 27-28. However, this does not mean the sins are equal. What Jesus was trying to get across to the Pharisees is that sin is still sin even if you only want to do the act, without actually carrying it out. The religious leaders of Jesus’ day taught that it was okay to think about anything you wanted to, as long as you did not act on those desires. Jesus is forcing them to realize that God judges a person’s thoughts as well as his actions. As He Himself said, [T]he things that come out of the mouth come from the heart, and these make a man ‘unclean.’ For out of the heart come evil thoughts, murder, adultery, sexual immorality, theft, false testimony, slander. These are what make a man ‘unclean’”—Jesus Christ, Matthew 15:18-20a. As He concluded earlier in Matthew’s gospel, The good man brings good things out of the good stored up in him, and the evil man brings evil things out of the evil stored up in him. But I tell you that men will have to give account on the day of judgment for every careless word they have spoken. For by your words you will be acquitted, and by your words you will be condemned”—Jesus Christ, Matthew 12:35-37. So, although Jesus said that lust and adultery are both sins, that does not mean they are equal. It is much worse to actually murder a person than it is to simply hate a person, even though they are both sins in God’s sight. There are degrees to sin. Some sins are worse than others. At the same time, in regard to both eternal consequences and salvation, all sins are the same.
So whether our sins are relatively small or great, they will place us in hell apart from God’s grace. The good news is that Jesus paid the penalty for our sins and the sins of the whole world at the Cross. If we will repent and turn to Jesus in faith, our sins will be forgiven, and we will receive the gift of eternal life as a result of a perfect Father seeing the perfection of His Son dwelling in us.