This chapter describes the division of the Levites for service in the tabernacle and then the Temple that David commissioned Solomon to build. David divided them by whether they descended from Gershon, Kohath, or Merari, the three sons of Levi. It reads partially as a genealogy. Moses evidently had several grandchildren that had progeny that later served for the family of Kohath.
Does God issue any commands?
None are given in this chapter but the commands concerning the offerings, shewbread, and things baked or fried are referenced. David made sure that each job had a Levite appointed to complete it.
Does God make any promises?
None are given.
Does this chapter teach anything about Jesus?
Everything was being done to keep the commands of God concerning the care of the tabernacle and the giving of offerings and sacrifices. Just as David did everything in his power to make sure God's commands were followed Jesus came to do the will of the Father by fulfilling the law.
Does this chapter teach anything about yet-future events?
David was preparing a place for the continual worship of God. Jesus went to prepare a place for us and will come again and receive us unto Himself so that where He is we may be also. Since Jesus continually worships the Father, that is where we will be also.