The best places have been mentioned--the Porcupines, Pictured Rocks, and Isle Royale. I'll second the Porkies and Pictured Rocks, and I'm sure I'd do the same for Isle Royale if I'd ever been there. By the way, the ferry for Isle Royale leaves from Copper Harbor. The Porkies are really special, I think. Just watch when you go to any of these places as mosquitoes, then black flies, and then deer flies can make you miserable. Call the rangers for bug info.
If you're looking for something in the northern lower peninsula, you can head to North or South Manitou Island, both part of Sleeping Bear Dunes National Lakeshore. South is served by two ferries a day, so there are day trippers there; on the other hand, North is served by only one ferry a day, so everyone there will be an overnighter. It's also a larger island. Good for short trips.
A bit further south, Nordhouse Dunes is a small area but nice for a short weekend trip to Lake Michigan.