Sure, you should be able to find a great apartment in the area, and maybe later on buy a house. Use the time of your apartment lease to work with local realtors to get ready to find a house.
Not sure if they're still around, but see if you can make contact with O'Farrell Realtors in Friendswood. They've always been very active in, and aware of, the local market and might have some things to look at...maybe you can skip the rental/lease thing altogether and go straight to home ownership.
I don't mean to sound like a homer, but Friendswood schools -- if you got kids now or are thinking about it -- are top notch. Also consider Pearland, 5 miles west of Friendswood, which 30 years ago was a real redneck dump, but I'm given to understand now is quite a nice, improved suburb of Houston. If it might be important to you, Pearland is just off SH 288, and has quick access to downtown Houston, Minute Maid Park, Toyota Center, Reliant Stadium and other cool parts of town.