The last four chapters of Hebrews is often the most read chapters of the letter. It is the most practical. The first section of the book is a "sermon" on the book of Leviticus, on the superiority of Jesus Christ as a priest. It is helpful to understand this letter as an ancient homily as opposed to a long letter written to some distant friends. The teacher who wrote this message was expounding the Scriptures in order to bring out the best in a community of Hebrew Christians. These believers ought to have been more mature in their understanding of the Old Testament in light of Jesus' ministry; they ought also to have been more active in their love to others while enduring discipline with less whining. It's a sermon with very modern implications.
Things of interest: chapter eleven focuses primarily on characters from the book of Genesis as examples of "faith". A good exercise, if you want to better understand faith, is to read chapter eleven, read the story of the character in Genesis, and then come back to the sermon in chapter eleven and ponder the meaning of the message. You can't understand the message of the NT if you don't understand the message of the OT.
Chapter twelve encourages Christians who know nothing of persecution to not whine when life gets very difficult. If you make foolish decisions, God will discipline (instruct, correct) you so that you grow up and stop making stupid choices. Thank God he helps you learn from your past (assuming you are willing to let him walk with you...though you still have to walk through the hard times you have created for yourself...otherwise you won't really learn anything).
Chapter thirteen is all about loving other people. The trademark of a Jesus-follower is their love for other people. Who do you love that is not part of your family? That is not part of your church? That is not like you? That does not like you? Love people like God loves you as demonstrated through the life and teachings of Jesus. This is a great reason for why one should read their Bible and work hard at seeking to understand it and live by it. God knows your world needs it.