Outdoor activity in Fort Worth proper is centered on the Trinity Trails. The 40 plus miles of trails, bike paths, and bike lanes let you go out for a bike ride, run, or even bring your horse for some fun.
Start by going for a run downtown at Trinity Park - go as far as you want on the trail, on either the soft surface or the concrete sidewalk. Turn around and head back to where you started - it's that easy. Not a lot of shade along the trails, but you'll find lots of others out there with you, running, biking, or just enjoying nature.
Rent a bike and go out and explore more of the trail system - biking from downtown out the Clear Fork of the Trinity to Pecan Valley Park by Lake Benbrook would be an excellent way to spend the day, especially if you took your time.
If your tastes run more to off-road trail running or mountain biking, try the Northshore Trail on Lake Grapevine.