A couple of people have mentioned
Fredericksburg (T82); it often gets mentioned for its on-field hotel and restaurant. Jay complained about his experience at the hotel...for anyone concerned about that, remember that _most_ people who visit Fredericksburg do so for reasons other than aviation and never go near the airport. A short cab ride into town gives you access to all the amenities the town has to offer, including plenty of other lodging choices. The original Nimitz museum has grown to become the
National Museum of the Pacific War, and it is an amazing place to visit. Much of the rest of the town is "touristy", but there's still plenty of good German food (and beer!). If you rent a car, touring the nearby Hill Country can also be fun.
Enchanted Rock is only a half-hour away, for instance. F-burg's a nice place to take your SO for a weekend getaway.
Boerne (5C1) is another nice place to visit for a weekend, but there's not as much to do in town, so keep it in mind if you're just looking to get away and relax.
San Antonio is within easy driving distance of Boerne...but if you want to visit SA, just fly into
Stinson (KSSF) and rent a car. The airport is right on the mission trail, so you can start your sightseeing immediately!
If you're in the eastern part of the state, consider
Woodville (09R), especially if you have younger kids.
Heritage Village and
The Pickett House Restaurant is open 360 days a year, and if you call 'em they'll send someone to the airport to pick you up.
In the SE part of the state,
Angleton/Lake Jackson ("Texas Gulf Coast Regional") airport (KLBX) has a great on-field restaurant,
The Crosswind Cafe. Fresh ingredients and fair prices (for an on-field place) and great homemade pies (baked fresh daily, they'll be long gone if you show up for a late supper!) make this a frequent evening weekday flight for me, though I haven't been in a couple of months. Call ahead, though, as their hours are somewhat odd.
Galveston was mentioned before for the
Lone Star Flight Museum. This is one of the best aviation museums in the southern part of the country...many of their planes are kept in (or are being restored to) flyable condition, including a B-17. Also nearby is
Moody Gardens, which has attractions for all ages, but can get kinda pricey if you try to hit several of them, especially as a family. There are a couple of restaurant options at MG even if you don't buy admission to the attractions. Schlitterbahn also has a location in Galveston adjacent to the field. All 3 places are within walking distance, but the FBO also will chauffeur you around in their golf carts, or you can call a cab or rent a car to get into town.
Near Dallas,
Stephenville has the
Hard 8 BBQ. It's not on-field, but they keep golf carts near the FBO that you can drive a short distance to the restaurant. This is a very popular destination for Dallas pilots, though, so don't count on the carts; it's not a bad walk if you have to do that, though there's little shade so it might get unpleasant at lunchtime in the summer.