Well, I'd go backpacking since it's not exactly a world-class climbing destination. I don't think mountain bikes are allowed because it's a wilderness. But you could bike around it on the access roads.
This TR reports a good 20-mile backpacking loop. For two nights, start at the Escarpment lot and camp here overlooking Lake of the Clouds a mile or so in (bring water); then the second night along Lake Superior -- Backpacker magazine says this site is the best place to go skinny dipping in the United States. (It will be cold, but you're a tough guy.) Then it's about four miles out the last day.
The whole park is likely to be muddy.
dwhike probably has more ideas.
Why on earth are you going to the UP for Memorial Day??