Some general notes:
3:15-27 – The end of the law and the reason for the law. The law was there to protect the Jews, keeping them on the path to God. Now, through Christ, we have a direct relationship with God.
4:21-31 – Abraham had two sons – one by a slave and one by his wife (a free woman). The slave child was born of human connivance and the free child was born by God’s promise. Isaiah wrote “The children of the barren woman now surpass the children of the chosen woman.” We are not children of the slave woman, but of the free woman.
5:2-3 – No more circumcision, no more law. Although, Jesus was circumcised, wasn’t he?
5:13-15 – Freedom from the law is not an excuse to do whatever you want. We are to use the freedom to serve each other in love. (See Matthew 22:39 or Leviticus 19:18)
5:19-21 – Trappings of the world that develop when not living in the spirit
6:4-5 – Carefully explore who you are and the work you have been given. Don’t be impressed with yourself. Don’t compare yourself with others.
6:11-13 – Those that follow the law do so to look good in front of others because they are too afraid to live by faith. They only selectively observe and boast of their success.