Since there is so much content, I'll focus on just two passages (giving the rest of you an opportunity to chime in on the sections that stood out for you):
1) The Beatitudes (5.3-10)
2) the closing two verses of chapter seven (7.28-29)
Jesus spells out the groups of people whom he calls "blessed" and what their rewards for that blessing will be. The people Jesus mentions in the Beatitudes are not the people who would often find themselves in seats of power, positions of authority, or with a great deal of influence within their community. The people Jesus calls blessed are not called blessed by the majority of society. They are the overlooked and the stepped-on.
In short, they are not the people we expect to be blessed, but Jesus rarely does what we expect.
The Closing Verses
As Jesus finishes three chapters of teaching, the crowds are "amazed at his teaching, because he taught as one who had authority, and not as their teachers of the law." The crowds sat slack-jawed wondering who this guy was who just taught them about life in community and living to a totally different standard. He was not the typical rabbi because he taught with "authority" - more so than other rabbis, scribes, and Pharisees of the day.
I wonder how many rabbis, scribes, and Pharisees who were there that day would have taken objection to the thoughts of the crowd?
What verse(s) stuck out the most to you in Matthew 5-7?