The more popular area for shopping is the Galleria area, Galleria Mall. Pretty big with plenty of places around the area too. A few attractions nearby as well, including an ice skating rink in the mall and the water wall near Williams Tower. Lighthound is in Pearland, south of Houston. I haven't yet been there, but would like to next time I am in the area.
In downtown, my favorite hangout is Minute Maid Park for a baseball game, then nearby is Flying Saucer which serves food but mostly known for their dozens and dozens and dozens of beers on tap. Good stuff and reasonably good food.
Also a cool place in downtown area is The Aquarium. Its a restaurant that has some ridiculous fish tanks and such, plus some rides such as ferris wheel and what not. They also have this restaurant at the Kemah Boardwalk, which is between Houston and Galveston. The boardwalk is definitely worth going to. Shops, food, rides, etc. It got torn up pretty bad during our last hurricane, but I am pretty sure its back to normal now.
I assume you have your own car while here, so its worth mentioning: many areas of Texas coast you can drive your car on. Some of these areas you do need a pass to do so. I always enjoy taking a little drive on Crystal Beach, though it has been a long time since I have done such.
Another area to "be" is The Woodlands, which is on the more North area of Houston on I-45. They have a variety of attractions and a little river with a boat running through the area that you can take a ride on for a few bucks. As you area heading in that direction, just south of The Woodlands is Old Town Spring.
Also worth some mentioning area the couple of state parks in Houston area. In south Houston you have Brazos Bend State Park which is popular for the alligators. Same with Huntsville State Park, which is likely a little too far north for you.
I have ~20 years experience in Houston, so if you have any other questions, let me/us know.
-Cameron