Texas has a bit of an "exceptionalism" thing going on - in most places in the US, you see the American flag flying outside peoples' houses. In Texas, you're more likely to see the "lone star" state flag. They regularly threaten secession from the union and think they're better than the rest of the US. They're kinda the "Brexit Britain" of the US.
Austin is a cool city - been there a few times. One of my favourite places over here. And it's very different from the Texas stereotype - young, progressive, geek-friendly, music everywhere, people bike and use public transport (they have a decent light rail system), and so on. Great food too - proper authentic Mexican food, Texas BBQ, good steaks, and surprisingly good Indian food!
As a Brit in the US, it's also nice because they have a good football team (Austin FC), Nando's and a decent British pub (run by Brits) that does a Full English and a good Sunday lunch!
But you don't have to go far out of Austin to be reminded where you are - miles and miles of nothing, megachurches, people in the biggest, most jacked-up pickup trucks you've ever seen (all the big car manufacturers make special Texas-only models!), people carrying guns, and so on.