We camp primarily in State Parks. The MN state parks are great, but for your size you will most likely need reservations if you are arriving for the weekend and they do not have full hook ups (just electric).
Someone mentioned Myre Big Island state park - this is a great park in Southern Minnesota. I guess if I were you I stop there for a quick overnight and then head up towards the north shore. Some parks that you wouldn't have a problem fitting in and are nice would be William O'Brien State Park (it's just north of Stillwater, a cute little touristy town with lots of antiques right on the St. Croix river and just south of Taylor's Falls where there's a great rock park and you can take a riverboat tour). Further north is Jay Cooke state park (just south of Duluth). But the most scenic part of our state is on the north shore and Temperance River is one of our favorite state parks. Big sites right on Lake Superior and close to lots of falls, hiking, and biking. There is also a city park in Grand Marais which is a great little town. The park doesn't have the greatest sites (you're pretty close together), but you can walk to town and the views of the lake are amazing. There are lots of little shops, great unique places to eat and it's where the Gunflint trail starts (if you're lucky you can see moose on the trail).
I'd probably loop over to Itasca State Park before heading back south. There is ALOT to see and do there and it's our favorite park to visit.
If you are looking for a park close to the Twin Cities Lebanon Hills Regional Park is great. It's close to the Mall of America, zoo, etc. It does fill up on weekends so you'd need a reservation there if you are staying then.