The best time of year to visit Anchorage for most outdoor activity is the summer - long days that really never get that dark around the solstice, cool weather, and the city likes to play outdoors. Of course, if you're into cross country skiing or ski-joring, winter would be a lot better.
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.
One of the best places to get away from the city is Turnagain Pass on the Seward Highway - you can keep going all the way down the Kenai Peninsula, but near the pass you have access to lots of good hiking trails and campgrounds without much in the way of development.
Really, any direction you drive out of Anchorage will be great once you get past the sprawl. Some great places to go are Seward and the Denali State Park within about 2 and a half hours to the south and the north.