San Antonio
San Antonio is a great all-year destination, as the weather never really gets too cold to be outdoors, but even in the summer there are green, leafy areas to hike or ride your bike in. As the temperatures rise, head to the rivers and lakes to spend a lazy afternoon.
A great way to spend a sunny saturday in summer is to wake up early in the morning and go for a run or mountain bike ride at Canyon Lake, about 30 miles north of the city. As the temps get high, head down to New Braunfels for barbecue at either Rudy's or Cooper's (can't go wrong with either) and then spend the afternoon tubing the Comal or the Guadalupe (out of Gruene, Texas). After tubing, if you have any energy left, go for an evening walk along the new stretch of the Riverwalk downtown.
On Sunday, sleep in late, get a Tex-Mex breakfast, and then bike the Mission Trail south from downtown through San Antonio Missions National Historic Park. There isn't a lot to see in the surrounding areas, but the missions themselves are some of the oldest buildings in the Southwest.
The New Braunfels/Gruene area is crowded and busy in summer, with tubing, kayaking, and the best water park in the world, Schlitterbahn, but it's for a reason - the spring fed rivers are beautiful and clear.
Government Canyon State Natural Area on the northwest side is undeveloped but has rocky, tough hiking, trail running and mountain biking.
Close to the airport, and much more mellow, is McAllister Park, with about 10 miles of hiking and biking trails. You'll need a map, because the trail system is a maze. Most of it is shaded, but the stretch up and over the dam is hot and exposed.
Fredericksburg is a great getaway from San Antonio. Look through the tourist shops, get something to eat at Rather Sweet Bakery, and then head up to Enchanted Rock State Natural Area.
Enchanted Rock State Natural Area is the best place to go for a quick hike up and down the mountain.
In the Hill Country State Natural Area, pretty much every trail ends up at an amazing overlook. Highly recommended is parking at the Equestrian Parking area and taking the trail to the right up to Sky Island. Take the trail all the way around Sky Island, being sure to stop at the bench.
In town, the hike/run at Eisenhower is pretty nice - the trails are short, but they are steep!
The Riverwalk right downtown near the Alamo and the RiverCenter mall is not exactly a bucolic destination. Head to the new part of the Riverwalk (Museum Reach, north of downtown) for a much more relaxed experience.