I rented a bike in Galveston summer before last. There were a couple of bike rental places along Seawall Blvd (the beach road). I can't remember what it cost, but it was by the hour, not by the day. The first thing you notice when you rent a bike is that you only get the bike, you're missing all the accessories such as water bottle, helmet, lock, tools, etc. Be sure to rent a lock or bring your own if you plan on stopping anywhere. Most of the rental bikes seemed to be single speed cruisers. I picked a geared bike but it was way out of adjustment.
The easiest first ride will be on the sidewalk along the beach side of Seawall. It's full of pedestrians though. Most of the downtown streets are very bikeable.