Hobby is the closer and smaller airport. If this option works for you, it's the easiest and best choice. Bush is a little bit further away and a bigger airport, so give yourself a little more time if you are flying in and out of here. Houston traffic is a consideration to keep in mind.
There are shuttles to and from the airports through Galveston Express and several hotels offer that service. If you opt to pick up a rental car in Galveston, we have an Enterprise on the island.