Don't know how many days you have or if you are renting a car or not.
Take one day and go to Whittier (car, bus, or train) and do the 27 Glacier Tour, it is very good, includes lunch.
As suggested, a drive to Seward to get a feel for how big Alaska really is, a lot to do in Seward, so that will be a long day. If you drive to Whittier for Glacier cruise you are about 20% of the way to Seward.
In Anchorage they have a Museum downtown by the convention center that is good for Alaska history. If you are there on a Saturday morning they have a farmers market downtown that is fun to walk through, I think it opens in May.
It may not be a warm as you expect in May! If you get a chance to go fishing, bring real warm clothes for being on the water in May. Fishing guides and trips are easy to find with information at all the hotels.
Food is easy to figure out, just avoid the chains. If you like Gin & tonics, there is a newer restaurant and bar south on old Seward at O'Malley that specializes in gin & tonics and has good food. I forgot the name. The best pizza is a place called Moose Toes (I think I got the name right?) on Old Seward just south of Tudor Road.
Have fun!