In my opinion, in Fairbanks you need a car. Cruisetour shuttle service isn't going to get you to most of the attractions. Fairbanks U has an excellent museum, botanical gardens, large animal research tour. Downtown has the Public Lands building, well worthwhile, plus the Ice Museum, again very interesting. There should be a shuttle to Pioneer Park, great exhibits, take the time to read the plaques on the buildlings. Very good salmon bake, but I ONLY go there when it opens. They have gotten so big with tour groups that they no longer exchange for clean plates at the salad bar. Gross contamination in my opinion. The Palace Theater has an excellent show. Ester Gold Camp is also worthwhile.
Denali Park, consider some of the park activities www.nps.gov/dena
ranger lead activites nightly, at Savage River, dog sled demo.
Copper Center, highpoint is McCarthey and Kennicott Mine, but my clear preference is flying in, very very rough slow going road otherwise. This is a great fishing area. The most scenic portion of the Richardson is Copper Center south to Valdez. Do you go this routing???
Cooper Landing, FISHING and more fishing, about the only activity in the area. There will be tours to Seward, if a shuttle available, Sealife Center, Exit Glacier, Seavey's sled dogs. Otherwise consider a Kenai fjords boat tour- of at least 6 HOURS ON THE WATER. Do not consider any shorter.
This portion of your trip is completely casual, never any time do you dress up.