This chapter contains the sermon on the mount, where Jesus outlines who will be rewarded and how. There are some nice things about the meek inheriting heaven and merciful people being shown mercy. Also he uses the expression salt of the earth (v. 13) which I did not know was biblical.
He assures everybody that he's not here to overturn the law, he's just fulfilling some prophecies. He in fact explicitly says that people who break the Old Testament laws will be punished in heaven. Note, too, that although Matthew is the first book in the New Testament, it was written about 80 years after the events it depicts ended, so clearly lots of OT laws were still being followed.
Of course none of this applies the Pharisees. They will never get into heaven. Who else is going to hell? Cursers and people who fight with their brothers. In fact, patching things up with your brother is so important, you should alter your sacrificing schedule to do it.
Other people not making the cut? Adulterers, which now include men who just look at women lustily have committed. In other words, everyone is going to hell.
How can we avoid sin? Plucking out our eyes and cutting off our hands. Yes, in the same chapter as that famous bit about the poor and downtrodden inheriting the earth, we get the other famous bit about cutting off the body parts that are causing you to sin. Wonderful.
Apparently in other gospels, Jesus will tell us that divorce is wrong and bad and never okay, but here he tells us that if a wife commits adultery, the husband is cleared for separation.
We're also instructed never to make oaths in heaven's name. So that locks the pearly gates to every woman who has ever been nine and tried to convince people that a lie is true by saying 'I swear to god!' Also, anyone who has ever been asked to tell 'The truth, the whole truth and nothing but the truth, so help you god.'
This chapter is extraordinarily muddled. First all the good people, then the bad, now we're back to good people again, who are supposed to turn the other cheek when someone hits them, and give their coats to people who steal them. In short Love your enemies (v. 44) which rings a little hollow after all that condemning to hell in the middle bit.