Of course there are the standards: Wendy's, Carl's Jr, McDonalds, Jack in the Box, Burger King, but I suspect that you can walk into one of these blindfolded and have no clue where in the U.S. you are.
As anyone who has come to visit us can attest, the best burgers in Bastrop come from a place called the Roadhouse. www.kendrasroadhouse.com When we first moved here we drove by the place and I assumed based on appearance that it was just another run down building. Boy, was I wrong! It sits at the confluence of 3 state highways, across the road from a state park and golf course and down the road from Lake Bastrop and the Colorado River. The place is packed on weekends and serves great burgers on either a sourdough or jalapeno cheese bun. Real milkshakes, too. And (this is definitely a Texas thing) they bring you another glass of whatever you are drinking in a to go cup to take with you!
Down the road is Sonic Burger, which I had never experienced before we moved here. We have been there once since we arrived in 2006. Need I say more?
Down the road from there is Whataburger - a Texas original Not sure how I feel about it, but it is open 24 hours and I like their ads. www.whataburger.com.
In Austin there is a relative newcomer that knocks the buns off all the fast food places: it is called Mighty Fine Burgers. There are 3 locations and I have been to 2 of them. The beef is all natural, the fries are cut there and they make lemonade and milkshakes from scratch. YUM! www.mightyfineburgers.com
There are numerous "only in Austin" places that we have not yet been to, but will keep you posted as we visit them. In the meantime, pass the napkins and how about that lemonade to go.