Anchorage is a great town to ride a comfort or hybrid bike around on the bike paths - two of the best are the Bird-to-Gird trail on the Seward Highway south of town, and the Tony Knowles Coastal Path near downtown.
Just outside of town on the Glenn Highway in Eagle River, you can visit Thunderbird Falls for a hike. In the same area, you can spend a whole day out at Eklutna Lake - do the bike and paddle package trip on the lake, where you ride a mountain bike out on the trail, but paddle back in a canoe or kayak.
Austin is probably the most laid-back outdoorsy city in the country. Nothing beats waking up in Austin with absolutely nothing to do. Best thing to do on a sunny Saturday would be to get down to Town Lake and go for a run/jog/walk with your dog around the lake (3, 4, 5, 7, and 10 mile loops) in the morning. Then go rent a mountain bike from Bicycle Sport Shop (or bring your own), and bike over to Jo's on 2nd Street for brunch outside.
Start out with a hike on the hilly trails at Ruffner Mountain Nature Center. Stop into the visitor center to learn more about wildlife in Alabama. Once you've worked up an appetite, head to Continental Bakery at 1909 Cahaba Road in English Village for some coffee. Take some food to go for a picnic, and go visit Vulcan Park to see the cast-iron statue of Vulcan on Red Mountain. If you have time, visit the museum at the park and go up to the Observation Balcony.
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.
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.
Start with a run around Audubon Park in Uptown between St. Charles Avenue and the Mississippi River. The road inside the park is 1.7 miles long, and is open for running and cycling without cars. To extend your run, head down the median strip of St. Charles where the street cars run (called neutral ground, locally) for a few miles and take a street car back to Audubon Park.
Spend it out by Lake Hefner! The site of the Redman Triathlon in September, the Lake Hefner trails are hard-surfaced and suitable for running or biking. You can ride your bike up on top of the Hefner dam for a great training ride, and then head out on the Redman bike course out of the city for longer rides.
Philadelphia has one of the best urban bicycle trails in the country, following the Schuylkill River out of the city to Valley Forge National Historical Park. Start at Boathouse Row along the Schuylkill, where you can rent a bike, and follow the path to Manayunk (a good turn-around point for shorter trips) or all the way up to Valley Forge. Up at Valley Forge, you can take a bicycle tour of the wintering grounds for George Washington's Continental Army.
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.