Been a San Antonio resident for nearly 20 years now so hopefully I can be of some assistance. The downtown area has been going through a revival in the past few years,mostly for the good. A few things:
1. Public transit is terrible here. The VIA bus system is useless.
2. North(pearl brewery area), south(southtown) and most recently the east side of downtown are definite IT areas to be. But, you're going to pay an arm and a leg in rent. They all offer walkable/bikable commutes in downtown.
3. SA is not a very bike-friendly city, but it's doable of you really wanted to.
4. West of downtown is like little Mexico. Nothing in a bad way, just not the area a typical recent grad would want to live
5. Unless you've got money for downtown, then you'll likely be moving further out and will need a car. You'll likely need a car and least for some things once again because of a lack of quality public trans and bike lanes.
Any other specific questions then feel free to ask. Hope that helps some!