Kincaid Park offers the easiest way to get deep in the woods right in town. This 1400-acre forest sits atop an old glacial moraine and offers one of America's top trail systems as well as Anchorage's largest moose population. Park at the Kincaid Outdoor Center or "chalet," (343-6397). If you want to test yourself, try the 6-mile Lekisch Loop; it's the hilliest competition-certified 10K in the country.
You'll also find miles and miles of singletrack biking trails for all abilities. Rent a bike from Downtown Bicycle Rental and explore for yourself!