The Minnesota Planetarium owns a portable planetarium, the ExploraDome. It's usually captivating students at local schools, but the ExploraDome also visits community events like the Minnesota State Fair and other local festivals.
St. Paul Planetarium is in Como Park, and is operated by St. Paul Public Schools. While it is mostly used by school groups, the St. Paul Planetarium has regular public programs and shows.
The University of Minnesota's Department of Astronomy opens to the public every Friday night during the spring and fall semesters. Once dark has fallen, a students and staff give a short presentation followed by stargazing with the University's telescopes on the roof of the Physics laboratory. Public nights are free to attend, but viewing is not possible if the weather is too cold or the sky isn't clear.
And during the summer, another U program, Universe in the Park, visits State Parks around the Twin Cities for free stargazing programs. Here's the schedule for Universe in the Park.
The Minnesota Astronomical Society is one of the largest astronomy clubs in the US. The MAS has regular "star parties" and operates their own observatory in Baylor Regional Park, near Norwood Young America. The public and those interested in joining the MAS are welcome at many of their events at locations around the Twin Cities.
For stargazing on your own, locations in Minneapolis and St. Paul have too much artificial light at night which makes it difficult or impossible to see faint objects in the sky. State and regional parks around the Twin Cities metro area, either in the suburbs or a little way out of town, are a good choice, and you can camp out and stay all night. Camping is available at state parks like Minnesota Valley, William O'Brian, Interstate and Afton. Several parks in the Three Rivers Parks District have campsites. Camping is also available at many other regional parks outside of the center of the Twin Cities.
And for anyone who doesn't think Astronomy is fun - students at Northfield's Carlton College turned their observatory into R2D2, a brilliant prank making the small Minnesota college famous for a few minutes on the internet.