Great idea. Alaska on your own is so much different than from the inside of a tour bus plus you can go where the busses don't. Particularly recommended is the Kenai Peninsula and the MatSu region, both easily accessible from Anchorage. Longer trips include Valdez (the terminus of the Alaska Pipeline) and Fairbanks (our 2nd largest city with a completely different flavor from Anchorage).

One of the best (and little known to visitors) one-day drives from Anchorage is to Palmer, Hatcher Pass and the Independence Mine State Historic Park. Awesome scenery and parts of the road are an adventure for those used to driving on a freeway (go easy in a passenger car).