For good rainbow fishing out of Anchorage you will have to do a little driving. Here are a couple of options:

  1. Drive down to Cooper Landing on the Kenai. It is ground zero for big trout. Check in with the fine folks at Alaska Troutfitters about getting a guided day on the water. The Kenai is a challenging one to fish if you don't have a boat. If you time is limited a guided day is money well spent.
  2. Drive north out of Anchorage and head for some of the great streams off the Parks Highway. Willow Creek, Montana, etc are all good trout rivers. They are much more fishable on foot. The rainbows are generally not as monstrous as Kenai bows, but there are some really nice ones. It might take some walking to leave the crowds behind but you can do it. 

Either way you go, you'll have some fun!