I took that route several years ago, but after several rainy days I stayed 2 days at Chena Hot Springs and then on my return trip also took the bus tour through Denali. The Denali was the best part, both for the mountain scenery and wildlife. In June, the days are long there, so you can start early and drive until late evening. If you leave Fairbanks at 3-4am, you should be able to squeeze in the Denali morning tour, and after the tour continue driving to Talkeetna.
The road is pretty good, without too much traffic. You may see moose also from the highway or near some settlements on your way. As to the official hiking trails on that route, most are quite long tours.
