I'm a native Texan but Minnesota is home now. If you're looking for things to see in Texas, head to the Hill Country. The smaller towns and cities are the best parts, but Austin and San Antonio are full of cool stuff. By Texas standards that's a short trip from DFW. If that's too far for you, don't bother going far from the hotel. Fort Worth is generally more chill than Dallas, but both have a lot to see and do.
After the those places, you gotta go for a good road trip to see much worthwhile: Lubbock, Big Bend, El Paso, Corpus Christi.
I don't recommend anywhere remotely close to Houston, Abilene, Midland, Odessa. These are all places you should only be if you're being paid to be there.
You can see areas similar to most of Northeast Texas minutes east of Dallas. A few minutes west of Fort Worth will give you a good idea about the inhabited parts of West Texas. Things get scrubby and sparce pretty fast out that way.
Big bonus points: there's a high concentration of South Indian people north of Dallas, relatively near the airport. If you aren't familiar with South Indian food, go seek it out. It's different from the North Indian food available in most US Indian restaurants.
Aside from that, you want to eat Tex Mex (not real Mexican food), any small town local BBQ place, and a decent chicken fried steak (they aren't all equal).
I generally recommend local restaurants, but if you're looking for chains: The best meal at Whataburger is breakfast, but it's gone downhill in the last 20 years. Not usually bad now, but not what it used not be. Braums has better burgers and better breakfast (fight me) if you get a good store. They are hit or miss though. People say Dairy Queen is a Texas Stop sign, but I think Sonic feels more "Texan."
