One of best things about living in Anchorage is its multiple stretches of undeveloped areas with paved and unpaved trails running through them. Many of the trails go through tunnels underneath roads, so for long periods of time there are no worries of motorized traffic. The chance for seeing moose is very high and scenery goes from small ponds, creeks, and spruce forests to open fields, parks, and neighborhoods. In the winter, people use these trails for cross country skiing. 11 miles of trail are lit up for winter use (and with only 5 hours of sunlight in the middle of winter, it's surely welcome).
If you are a visitor, and you have an afternoon free, rent a bike and check these trails out. Just make sure you eventually head towards and ride on the Coastal Trail for the best biking in town.
It is highly likely that you will encounter moose on the trail (and I mean ON! the trail). Be careful riding fast when you can't see far ahead of you. If a moose is off to the side you can usually pass without impunity, but don't linger. If you see a cow with her calves, use extreme caution! When in doubt, turn back. Please go to my Education page for more information.
Also be aware that there will be many people on the trail using it for many different reasons (running, dog walking, biking, etc.). Stay to the right and respect everyone's space.
Get the map, Anchorage and Vicinity (Alaska Road and Recreation Maps, Publisher) for details or, Anchorage's Best Trails Map - A brochure of all the biking, hiking, and skiing trails in the city can be purchased at the Anchorage Convention Center & Visitors Bureau, "the Log Cabin", at 4th and F.
You can rent bicycles from many places downtown and midtown. There's a great shop at 4th and D in the Ship Creek Plaza.
A note about smart urban planning: many of Anchorage's public schools are connected to these trails. If you lived here, your child could walk to school and not have to deal with dangerous traffic.