Fort Worth has become quite bike friendly of late. Lots of trails in and around town. They even have a bikeshare program in the downtown area. (It helps that the mayor is an avid road biker)
As for attractions: the zoo is very good. Downtown is pretty vibrant after the workday is over. Lots of shops and restaurants. It is a bit 'touristy' but the Stockyards are pretty cool.
I live in Dallas and would not suggest it for any type of bike trip.
That is all.