There is a small "old time" bike shop downtown, just three blocks off Houston street (where many of the tourist hotels are located) that rents beach cruiser type bikes. The staff at a downtown hotel can give you directions, it is right next to the Riverwalk.
Downtown is not the best area for riding though. The streets have lots of traffic. Most of the tourist attractions are on the Riverwalk, and bikes are not allowed on the walk (too narrow and too many people). There is an historic area just south of downtown that would be pleasant for riding, as would the parks around the zoo, five miles north of downtown.