Shakespeare's Tempest is an interesting play; it's one of the few that is hard to categorize, and it is one that seems to be completely without background source material, unlike most of his works, which drew from myth and older stories.
As he usually does in his plays, Shakespeare explores multiple themes through multiple plotlines and character development. Some of the themes in Tempest are:
1. Kinship and relationships as expressed through Prospero's relationship with his brother, his daughter, and his servants.
2. Uses and abuses of power, again as expressed through Prospero, who has traded the power of governmental authority for the power of words, through books and magic. Shakespeare also addresses how Prospero's power is expressed differently through his servants Ariel and Caliban and his power over the elements.
3. The consequences of knowledge as expressed through the use of magic, which Prospero gives up at the end.
4. The idea of fate vs. human intervention, since though Prospero seems to be able to control everything, he cannot control the way people's emotions and actions will affect the outcome of his machinations.
There are other themes that you may find in your own reading, but these should be enough to get you started.