I'm a San Antonio "native", in that I've lived here most of my life. I did leave for about five years and came back a year ago for nursing school. Things changed a lot (more then I could ever explain - thousands upon thousands of house where it used to be nothing but empty fields for a dozen miles - all over the city, not just in a few areas) while I was gone, but I'll try to offer some housing advice best as I can.
The neighborhood directly around UTHSC, especially between UTHSC and I-10 can be so-so. There are some REALLY bad neighborhoods there, especially off of Wurzbach, but you'll know them when you see them. There are some really good neighborhoods there too.
South of the school is pretty good. Culebra @ 1604 is a good, cheap place to get a house, especially inside the loop (1604). Marbach outside the 410 (loop) is good, and cheap. South of Marbach is a no-go zone in my book.
Pretty much anything NW of the school is good. The farther north, the better.
The south side and east side are no good. The east side is too much of a commute, and there's not much out there but crime (for the most part). The south side is bad. I lived there for many years and I can't, in good faith recommend it (although I had no problems, I know a lot of people who did).
North of the airport on 281 is excellent. Right now I live at 281 @ loop 1604 and it's probably the best area in town to live in. Virtually no crime, it's all brand new, lots of young successful professionals raising families. Outside the loop almost every single neighborhood has a private pool, and there is a ton of semi-private parks (they're open to all, but inside the neighborhoods). The rent is steep, but if you can get 4 to 5 people together, you can rent an awesome <8yo two story house for <$400/each, and about half these neighborhoods are gated. The drive isn't the best (would take approximately 30-45min to get in every morning), but they just changed the main highway to a new style (it's a Super Highway now), which means it has a ton of confusing traffic lights, but it's cut drive time in half, and then some.
As far as what to do, just depends on where you live. There's not really one spot to go to party or shop or get food. San Antonio is a massive city (size wise it may be the biggest I've ever seen in the country) and, as such, there are pockets of EVERYTHING just all over the city. Movie theater, mall, every kind of shopping/food imagineable, all in a cluster here and there dotting the city. Downtown is fun to do every now and again, but the natives seldom go down town more then every other month or so during the night (parking is ridiculous, it's packed, and lots of riff-raff). During the day time, downtown is amazing and the River Walk is something I still haven't got tired of. However, whenever they're having a festival or get together down town at night, it's more than worth going.
In the end, here's a quick break down laying out the city:
Downtown is nice, but expensive.
West side (which encompasses vastly different areas depending on who you ask) is priced good to rent and generally very safe.
South side extremely cheap, but not very safe. Long commute too UTHSC.
East side very cheap, not very safe at all. Longer commute to UTHSC.
Northside, decent prices, very safe, decent commute.
UTHSC is generally considered to be on the West or, more commonly, the NW side of town. It's between loop 410 (which is the "inside" loop going around the city) and loop 1604 (which is the "outside" loop). It has easy access from HWY 410 and IH-10, which means commuting from the north or west side is a snap.
Hop on Google maps and type in San Antonio and this will all make a ton more sense. If you want, feel free to send me a message asking about specific areas, and if I know them, I'll tell you what I know. I used to own a property management business in San Antonio about 6 years ago and I managed a lot of properties on the West, South and East sides of the city.
And finally, if you enjoy golfing, welcome to paradise. Go to one of the five municipal courses and buy a year long players club membership for $50 bucks and get 6 free rounds (you have to rent a cart for $12 is the only catch) at three of the different good munies (2/per course), as well as discounts at all of the munies, and LOTS of free group lessons. Were it not for school (nearly done with my nursing program), I'd be golfing at least once a week for less than $50 bucks a round (that's green fees, cart, and a couple drinks from the pro shop).