Well, you've got quite a few deer in your neck of the woods, but to get after them, here's what you'll need to do.
First, get in contact with the folks at Texas Parks and Wildlife to make sure I didn't skip anything, and to keep it legal. They'll also be able to direct you on how to get your Hunter's Safety certificate - It's manditory if you were born after the grandfather date, which I don't know off hand. check
http://www.tpwd.state.tx.us/
Second, go to Walmart or Academy, or just about anywhere out there really, and buy a hunting license
Third, if there's public land to hunt near Austin I haven't run across it. Most likely you'll have to either get on a deer lease, or go on a day hunt. A deer lease is ussually for the hunting season, or whole year, depending on the land owner, and will run in the neighborhood of $1000 per gun, some a little less, many a lot more. a Day Hunt can run $75-$150 a day, more or less.
Generally, on a day hunt in your area, a "guide" will drop you off at a deer stand, ussually overlooking a feeder (don't want to argue the ethics of this, that's just how it ussually happens), and instruct you not to leave the stand. This is probably to limit their liability, but I've only found one day lease that would let me walk, and he doesn't do hunts anymore. Anyway, you'll sit and wait, and in all probability see several deer.
You'll have a pretty hard time finding a Deer lease available this time of year, but day hunts are pretty easy to find, especially if you can work it on a week day. Anywhere to the west of Austin should be great, especially around Kerrvile, Llano, and Junction. They aren't huge, but plentifull. Personally, I hunt outside of Eden, west of Brady, but unfortunately our lease doesn't allow guests. There are TONS of deer in your area (drive out to Llano one night when the moon is full -slow down, and use highbeams!).
Now the best way to hunt out there is to be invited to hunt on someones ranch, and if you figure out how to do this often, tell me how.
Anyway, If you are wanting to get your feet wet, for the smallest investment, the day hunt is the way to go, but you'll have more rules to follow, and won't know as much about the crowd hunting with you as you will on a full-time lease.
Also, don't discount stand hunting. I enjoy getting out and walking around, and have gotten a real nice buck and couple of does this way, but I always see more when I sit down, be quiet, and wait. My personal hunting style(when I have the room to do it) is to walk, sit for an hour, walk, sit, walk, sit.... you get the picture.
Anyway, if you have more questions, put 'em here, There's several other Texas hunter's about