You have about four to six weeks to get down here before it gets really hot. As far as a tourist destination, San Antonio is pretty much the default in Texas for traditional tourist stuff. There is the Alamo, Riverwalk, El Mercado, Mission trail and amusement parks (Sea World and Six Flags). The San Antonio Fiesta is starting this week and runs to April 30. There are lots of events, but also lots of huge crowds and hotels are expensive and fill up fast. Even when it isn't Fiesta week, downtown around the Riverwalk gets pretty packed with people on the weekend. So, if you like hustle and bustle of big crowds, San Antonio would be fun on the weekend. If you want to be somewhere that is not as much of a tourist trap, keep reading.
The rest of Texas is not as much of a traditional tourist destination, although Austin is sort of becoming one. Austin is basically Portland, OR, but with sun and not as many homeless people. Downtown Austin is just north of Lady Bird Lake, which is a small lake made from a dam on the Colorado River. You can take out kayaks, canoes, paddle boards and pedal boats on the lake. The Harry Ransom center is kind of a Texas version of the Smithsonian and is pretty neat. But the real fun in Austin is going out to the Hill Country. Within about a 1-2 hr drive, you can get to some really great state parks. The best one is Enchanted Rock. But be ware that you have to get there early on the weekend as park capacity is limited. There is also Pedernales State Park, Colorado Bend State Park and Longhorn Caverns State Park. Fredericksburg is a touristy small town near Enchanted Rock.
Dallas and Houston just aren't tourist towns. They have nice museums, a few interesting parks and neighborhoods and lots of great restaurants. But if you want to have a fun weekend in late spring, the Hill Country (Austin/San Antonio) is the best. Save a visit to Dallas and Houston if you are narrowing in on a college choice.
If you put off your trip until later in the fall or early winter, take a look at Big Bend. Right now it is getting too hot to really enjoy Big Bend with temps pushing 100 in Terlingua. But when the weather is good in the spring and fall, it is a pretty magical place.
As for nursing schools, there are tons. For the traditional college experience, UT Austin has everything. For a more focused program, take a look at specialty nursing programs in the Med Center in Houston. But there are great programs all over the state. Texas Woman's University in Denton (just north of Dallas) has an excellent program. Thanks to a big music program at UNT next door, Denton has a great live music scene and is a fun little town for college kids.