There's a place called Thunderbird Falls near your friend where you hike back a ways to see the falls. Definitely see Denali park. Earthquake park in Anchorage might be interesting.
Be sure to see Portage glacier--take Seward Highway south of Anchorage a ways. If you can make it to Seward you can take a half-day "cruise" around the bay and see lots and lots of eagles, puffins, sea lions and such. The Anchorage zoo is a nice visit; all of the animals are either injured or overflow from other zoos; they try to rehabilitate the animals and return them to the wild, so the animals rotate a lot.
You may be there while the state fair is going, and it will be in your area. Definitely go there if it's on. Browse the agriculture shows and see huge veggies.
We used to take a trip on the Whittier ferry. I was a kid, so my details may be fuzzed up, but we drove the car onto the train carrier towards Portage glacier from Anchorage then rode the train to Whittier and put the car on the ferry. It goes along a lot of glaciers and they used to let kids sound the horn and knock some ice off. We saw killer whales on our ferry trip, but that's fairly rare. I believe the ferry winds up its trip in Seward where you can drive back to Palmer.
Homer is a bit far, but as a kid I enjoyed seeing the 200-lb halibut being brought in.
Somewhere in the vicinity of Thunderbird falls is a series of beaver dams; I don't think they are marked, but if you know how to find them I found it enjoyable to hike back there and watch the beavers toil away.
The big ski areas are Alyeska (S from Anchorage) and Arctic Valley (N of Anchorage). I never really cared to go there when there's no snow, but I understand they run a lift or two, and you can ride up to the top and look around and ride back down. Come to think of it Alyeska has a gondola now from the new (to me) resort hotel to the restaurant up the mountain.
If you're active, there is a bike trail that runs along the coastline from downtown to a park. I saw several moose on that ride, but you'll probably see moose everywhere.
Do hike around one of the creeks or rivers; Bird creek is nice and near Anchorage, but there are rivers freakin' everywhere.
Well, that's a start, anyway. It's hard to go wrong in Alaska. The Arctic Roadrunner has good burgers in Anchorage, and Gwenny's is a nice place to eat.
Take your camera and lots of film/storage.