First, welcome to the forum. In Ft. Worth, there is a bike path that runs from near downtown in Trinity Park, all the way to near 820 and Bryan Irvin. Also, riding around the cultural district is a fun and generally safe thing to do, unless the Stock Show is going on. Just get out and ride, you will be surprised how many people will watch out for you as long as you keep your wits about you and ride safely.
Good luck and happy riding.