Dallas can be absolutely beautiful in the spring with the flowers and moderate temperatures, but gets extremely hot in the summer. Winters are constantly changing - it can be 70 one day and sleeting the next. Fall is usually nice weather, especially in November and December.
Do what everyone does to exercise in Dallas and head to White Rock Lake for running or bicycling. Heck, bring your dog and let him play in the off-leash dog park (fenced in) next to the trail.
Cedar Hill State Park is not too far from downtown, but you will get a little bit of the Hill Country. For piney woods and some great mountain biking, go a little further out to Tyler State Park.
You can also go to Fort Worth to explore their outdoor activities.
Dinosaur Valley State Park and the Granbury region southwest of Dallas is a great area to spend a weekend hiking, mountain biking, or relaxing by the lake.