Anchorage is very bicycle friendly. For the most part you can ride on the sidewalks. In theory you are not supposed to ride on the sidewalks in the downtown business core, however, I have found that if you travel along at a pedestrian rate, no one worries about it.

There are 120 miles of paved shared use trails in the city. The nicest is the Tony Knowles Coastal Trail (which does start in town), however, there are others that are nearly as nice, running through forests and along creeks and rivers. If going in a straight line is not a requirement, you can get around most of Anchorage on bike trails with limited actual on road riding. Just keep in mind that the trails are multi-use and that the slowest moving party has the right of way.

As to the Downtown Bicycle Rental, as far as I know they are reputable. As I live here I have my own bike and so don't need to rent. I have travelled by their shop many times and they are conveniently located in downtown and are only a few blocks from the coastal trail. There is another rental spot in town, Pablo's Bicycle Rental which is a little smaller than Downtown, however seem to have a decent selection and are just a few hundred feet from the coastal trail.

As to best routes, there is are maps of the local bicycle trails that you can get wherever you rent. Depending on how energetic you are feeling, there are bike trails all the way to Eagle River to the North East and a large portion of the road South to Girdwood now has paved bike paths as well. I generally do my riding in town, commuting to work and then just for fun on weekends or going into town in the evening. Nothing is very far away in Anchorage so unless it is the depths of Winter, a bike works well. Next to the coastal trail, my favorite trail is the one along Chester Creek. It connects to the coastal trail a mile or two out of town, depending where you begin, and then heads inland past Westchester Lagoon and along Chester Creek for some miles. It branches off into various neighborhoods and eventually into the UAA campus near Goose Lake. Another nice ride is along the Campbell Creek.

One thing to keep in mind is that the wildlife is sometimes a danger. Moose encounters are common. Mostly they are okay if you don't approach them, but they can be aggressive if they have babies around. Bear encounters have been happening more frequently in recent years, especially on trails at the Eastern edge of town bordering Chugach park. There were a couple of bear maulings along the Eastern trails last year. On the other hand, we are in Alaska and wildlife is just one of the hazards of living here, along with volcanoes, earthquakes, blizzards, tsunamis, etc., etc.

You can see part of the Anchorage trail system here:

http://www.muni.org/iceimages/parks/TrailwatchMap.pdf

Hope you have a great trip!