I'm also looking at moving to Austin, so I've been doing some research. This'll be what I've found out, along with some general wisdom about moving between states.
1. Jobs are easy to find
Businesses need to hire a lot of people, especially when they're just starting up. Luckily, Austin has been growing a lot in the past few years, so new businesses are coming in to the region pretty quickly. As far as entry level jobs go, you'll need to compete with the rest of the unemployed-but-looking population. As such, jobs are easy to find, but they can be harder to get
Jobs are easier to find if you have a solid resume. Not sure if you're currently employed, but it looks a lot better if you have at least one year with each company you work at. Shows commitment. If there's been a gap between your last job and now, try to pad your resume with volunteer work related to the field you want to go in to, if it's possible.
2. Austin is a growing city
Like I said, Austin has really been growing a lot in the past few years, which means that a lot of people are moving into the region. Land prices are going up, and so is the cost of living. There are a few places in Austin where land values are still kinda low, but you'll generally be farther away from downtown, and I hear that commuting on Austin highways is a complete nightmare.
Before you even think about moving, you need to figure out if your new job will pay for your cost of living. You seriously do not want to be homeless.
Next, when you decide to move to Austin, if you decide to rent, find an apartment locator. They'll help you find a place to stay, and the extras you want in your home, while keeping your budget in mind.
3. Be smart before you move
If you're moving out for the first time, make sure you have a safety net. It's easier if your parents decide to back you, but otherwise, you need to be hella careful. I think I heard that you should have at least $4000 saved before you move, just in case. You'll also need to keep those extra fees in mind, whether it's to pay for a moving van, plane tickets, whatever.
Moving to a new state is hard- there are a lot of factors at play. Don't rush this shit, it's a serious decision to make.
tl;dr- you can find an apartment for like $700/m depending on where you live. Keep transportation fees in mind, 'cause uber gets expensive.
No one is gonna hand you a job if you walk up and say "I just moved in can I work here." Make sure your resume is water tight, that you can be reliable, whatever, before you move. It looks like HEB pays about $9.50 for cashiers, so that's not too bad for entry level work.
And hell, talk to your parents about it. Chances are, they'll be able to give you good advice. Maybe they'll even be able to help you out. You won't know unless you ask.
I found most of my info on /r/Austin, it's a good place to start, with advice from people who already live there.