1) Go to the art museums in Ft Worth, specifically the Kimble and the Modern. These are some of the best collections in the country and are for some reason often overlooked
2) If you like jazz there are some good jazz bars in Ft Worth. I can't remember the names of them....but some google-fu should find you some stuff. For some reason its something Ft Worth is famous for.
3) Cousin's BBQ is good, another to check out is Angelo's (my personal fav for the area).
4) Ft Worth has a pretty great zoo, and a nice Botanical Garden right next to it.
5) If you are a big planner, you may want to try and catch a play/opera/symphony at the Bass Music Hall. The Bass family is like the Ft Worth Rockefellers, they built all sorts of cool stuff and the Bass Music Hall is very well built.
6) Or, if you're bringing kids, try to catch a show at Casa Manana. It's a strange theatre that is more kid friendly.
7) If you have the time you may want to check out either the Amon Carter Museum, which is a kind of strange cowboy/Texas museum, or the Stockyards, which is basically just a tourist trap.
8) You'll notice I haven't mentioned Dallas. Really no reason to go there, and plenty of reasons not to.