Austin has four seasons, although locals will tell you it doesn't. Summer can get up into the 100's, but is typically in the 90's and humid. Go running or bicycling in the morning or evening when the sun dies off – Austin doesn't have as big of a heat island effect as other cities. Austin also has plenty of shaded trails. Best thing to do when it is real hot out is get into the water – the water in Barton Springs Pool is 68 degrees year round. Spring and fall should be nice enough weather for anything you want to do outdoors, as long as it isn't raining. Winter can be erratic in Central Texas – the nights can drop into the 30's and 40's, and the days can be anywhere from 40 to 70. It only snows once every couple of years, so you can leave the snowshoes at home.
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. Town Lake trail is too crowded with runners and walkers for safe bicycling on weekends, so I would go ahead and ride the bike up the Shoal Creek hike and bike trail from Town Lake to 38th street, where the trail ends (about 4 miles each way) You can keep riding your bike up Shoal Creek Blvd from 38th Street to US 183 (10 mile round trip), for Austin's best on-street bicycling. Break your trip at Upper Crust bakery on 45th street and Burnet (take Shoal Creek Blvd north from 38th Street, then take a right on 46th street, go about 6 blocks to Burnet road (busy major road), and then go down the sidewalk to the bakery (closed at 1pm on Sundays though). They have outside seating and plenty of bicyclists.
Ride your bike back downtown to Town Lake, and either cruise it back to where you rented it, or put it back in your car. A great way to spend the afternoon is to go canoeing or kayaking on Town Lake – I prefer sit-in kayaks, which you can rent from the Rowing Dock on Town Lake for $10/hour. The Rowing Dock is confusing to find – take Barton Springs Road to Stratford Drive, drive under the Mopac/Loop 1 Expressway, and it's on the right. You can kayak from the Rowing Dock upstream on the Colorado River/Town Lake/Lady Bird Lake to Red Bud Island and watch the dogs play in the water. If you like canoes better, you can rent them from the concession in Zilker Park below Barton Springs. After kayaking, you might want to dry out by going for a hike in Barton Creek. You can hike in from the trailhead in Zilker Park, which is convenient if you are down at the lake, but the city made a huge mess of the trail for a sewer project. It's not a bad hike, but the further up you go, the prettier it gets. The trail is 7 miles long (one way), so you've got plenty to do. You might want to only go as far as the Spyglass entrance, (1 mile from Zilker). Walk up the entrance trail to the TacoDeli (closes way too early on weekends) for some pretty good tacos with outside seating.
If you are dry, and there's any sun left in the day, you'll be right at Barton Springs Pool, Austin's gem and one of a kind natural wonder. No other city in the US has a massive, clean, spring fed pool that you can swim in downtown(San Antonio doesn't let you swim in theirs). The spring's water quality has been a point of contention between developers who wanted to pave most of Southwest Austin for tract homes and strip malls, and environmentalists who seized upon the cause of the endangered Barton Springs salamander to block growth and preserve Barton Springs. Enjoy Barton Springs, and educate yourself on the underlying Edwards Aquifer at the small Splash! Museum in the building.
After all of this running, biking, swimming, hiking, and kayaking (well, ok you probably didn't do all of those!) you're probably getting hungry for dinner. The touristy restaurant row on Barton Springs Road right by Zilker Park has some good options, but can get crowded – Chuy's is classic Tex Mex, Shady Grove is chicken fried steak and burgers with a trailer park vibe (for real), and Uncle Billy's Brewpub has good barbecue and on-premises beer, among others. If you have a car or a real love of the Austin bus system, grab your dog and go to Freddy's on South 1st street (about a mile south of the river). Good margaritas, good burgers, friendly people, lots of dogs. My only complaints are a lack of parking and for some unexplainable reason, no draft beer.
Austin is unlike a lot of other cities in that a lot of outdoorsy active opportunities are concentrated in the city limits, because of a lack of public land in Texas.
That said, outside Austin are some great parks within an hour or so drive. For piney woods hiking, Bastrop State Park is unbeatable with 12 miles of soft, shaded trails. No major hills or stream crossings, but you can't bring your mountain bike.
For bike riding through pine trees, you will have to go to Rocky Hill Ranch in Smithville, which offers camping and a "saloon" with beer and burgers.
Another great place to go mountain biking is Bluff Creek Ranch in Warda, owned and operated by some of the friendliest people you'll meet, and a great hideaway for camping. Both Rocky Hill Ranch and Bluff Creek Ranch offer up mountain bike and trail running races throughout the year.
After a good rain storm, Pedernales Falls State park, an hour west of the city on Highway 290, will have some impressive waterfalls in the channel of the Pedernales River. If the river is placid, you can hike out to the series of waterfalls carved into the rock. Don't expect Niagara Falls, but Pedernales Falls has its own Texas beauty that is unlike anything else during a flash flood. The park also offers up 25 miles of hiking trails, camping, and one of the most relaxing places to sit in the water in Texas at the swimming beach.
The best weekend getaways in Austin are to the west, out in the Hill Country. The standouts are Enchanted Rock State Natural Area outside Fredericksburg and Lost Maples State Natural Area out towards Vanderpool. The town of Fredericksburg is great if you are shopping for knick-knacks, but otherwise doesn't have much to offer except touristy German food and beer. Prepare for mobs on any nice weekend, as Fredericksburg is an easy drive from both Austin and San Antonio, and listed in tons of travel magazines. Enchanted Rock is the best hill climb in Central Texas, and is really unlike anything else you may have seen, as it is the second largest exposed batholith in the United States after Stone Mountain in Georgia. There are about six miles of trails that circulate through the park, some of which are easy, and some of which are quite technical. If you make it out to Enchanted Rock, drive north from the park to Llano to Cooper's BBQ in Llano for a real treat - open air grilled barbecue. The pork chops are especially good.
Lost Maples is all together different. It attracts the most visitors in the fall, when Texans come to see the leaves change color on the maple trees in the park. The maple trees survive at the bottom of the canyons, a reminder of a time when Texas was cooler and had a more temperate climate. Another place to see them is McKittrick Canyon in Guadalupe Mountains National Park. The park has ten miles of rugged trails, with outstanding views over the Hill Country. Camping is comfortable, but the campground is small, so reserve a spot ahead of time. There are showers to clean up with if you camp. The surrounding roads are perfect for road cycling, and there are charity rides through that part of the Hill Country.
Austin is hilly, but the biggest elevation difference you are going to find is only a few hundred feet. The classic Austin "mountain" is the staircase up to Mount Bonnell, but other great places to hike hills include Wild Basin Preserve and the Scottish Woods trail entrance to the Barton Creek Greenbelt. Both of these, you will start at the top and hike down, then hike back up. Austin doesn't really have any great climbing hikes - the best in the area is Enchanted Rock, above, other than that, you are looking at a 500 mile road trip to either Big Bend National Park or Guadalupe Mountains National Park.
Running, walking, or biking the entire Lady Bird Lake loop isn't as great as it could be, because a two-mile or so chunk of the trail (the eastern part of the south side) is actually on sidewalk on busy roads – not very scenic. Best to stick to the north shore between Congress Avenue and Pleasant Valley, though there is some really nice underused trail on the south shore just west of Pleasant Valley.
Lake Austin could be really nice, but it suffers from really poor public access. Unlike Lady Bird Lake, most of the shore is private development, and there are few parks. The parks that are on the lake are smaller, expensive, or very remote. The much larger Lake Travis has much better public access than Lake Austin. However, Commons Ford Park on Lake Austin is one of the most beautiful places in Texas.
The Waller Creek greenbelt is on maps, but not recommended, as it is not particularly maintainted, and attracts a pretty edgy element. The Shoal Creek greenbelt used to be similar, but has been cleaned up a lot in recent years with all of the development on the west side of downtown.
Commons Ford Park, Westcave Preserve, St. Edwards Park, pretty much anything with running water on a warm Texas day.