Take I-10 WEST from Houston all the way across to U.S. 285 to PECOS and SANDERSON off EXIT 257 in Ft. Stockton. To bypass SAN ANTONIO, take I-410 NORTH (& WEST). (EXIT 581 off I-10 to get onto...
From most parts of Houston, take the most direct route to the Sam Houston Tollway, then take the tollway west (if you are on the north side) or north (if you are on the west side) to Texas State...
I've made that trip a couple times. It's about an hour by car from downtown Houston to Galveston. There is a long bridge over Galveston Bay that gets you to the island of Galveston.