Youll most likely want a car if you want to do anything other than eat at a chain restaurant and stay in an airport hotel... It takes about 35 minutes to get to the Galleria (outside rush hours). The Galleria has a Westin, an Aloft, a Marriott, a Hilton, a Doubletree, lots of options, its the main high end mall in Houston, 3 levels, awesome restaurants, ice skating rinks... totally walkable from any of the Galleria hotels and can keep you more than entertained for one day...

There is always Galveston if you arent traveling during rush hours, its a bit of a drive, but being from TX, distance is relative... haha.

Let us know what kinds of things you like, and what times you arrive-depart and we can give better advice.