You will find that Texans get around in cars, the freeway system is extensive! "Metro" in Houston offers buses and trains through out the area. "www.ridemetro.org" should provide you information on routes, schedules, etc. Metro's transportation system is not anywhere near what you have in the UK. Pickup stations are open air with a small cover.
The Cypress Outlet Mall that you referred to is about 70 miles from Galveston, about 35 miles from downtown Houston.
I am not sure if there is a car rental facility in Galveston, suggest you contact the Galveston Chamber of Commerce or Galveston Visitors Center.