The Capital city, St. Paul, is about 10 to 15 minutes from the Mall of America and International Airport. I highly recommend visting; The State Capitol on Universtiy Ave. and Rice Sts. It's one of the most architecturally touted capital buildings in the country. They offer free tours several times a day, and the grounds and gardens are lovely.
Less than a mile from the capitol on John Ireland Blvd. and Concordia Ave. is the Minnesota History Museum. It's really fun and you can learn alot about the history of the founding of the state of Minnesota and the people who settled it. They have a theater and often free plays, kid crafts and activities, Costumed tour guides, a very nice restaurant and more.
About two blocks from the History Center is the world famous Cathedral of St. Paul. It is the 3rd largest Cathedral built in the United States and is breathtakingly beautiful, architecturally designed in the style of the Classical Renaissance era (like the Vatican in Rome). They too offer free guided tours daily.
Just 5 minutes away, in downtown St. Paul, there is the Minnesota Children's Museum (which is a real hoot for the young ones...the theme changes throughout the year in their main exhibit, but everything is interactive and hands on...sort of like a fantastic giant park indoors. They have alot of craft activities and storytelling too. Third Sundays are admission free!
Como Park in St. Paul, about 5-10 minutes North of the State Capital off Lexington Avenue, is an absolute gem and kid favorite. Admission to the Zoo is free..and while small compared to most zoos, is very impressive. They have a primate house, a fantastically wonderful Tropical wildlife exhibit complete with exotic tropical fish, plants and free flying tropical birds, an Anaconda, a cool sloth bear, ant exhibit and more, a very popular Seal show, Polar exhibit including Polar bears, penguins and more. New this year is my absolute favorite...a Butterfly house. You go inside a screened terrarium filled with free-flying exotic and common butterlfies and moths. It's quite an amazing experience walking into the "catepillar" hut and being greeted literally by a delicate and colorful swarm of friendly butterflies. My children did not want to leave. Seeing them up close, and sometimes having one land on their shoulders and heads was the biggest thrill of all!
There is also a beautiful green house on the grounds with fountains, ponds, and beautiful and rare plants..and an outdoor Zen Garden. There is also nearby Como Lake for which the zoo and park are named, and you can stroll the trail, rent a paddle boat, or catch a free music concert, or play a game of golf on one of the state's more challenging courses.
There is also an adjoining amusement park at Como with rides aimed at kids from 3 to 12. I personally enjoy this park more than the much talked about Mall of America-Nickolodean(sp) park. It is very affordable, and the kids love it. One child could ride everything for under $20. And there are alot of rides to choose from including a firefighter ride where kids can put out pretend fires with real water hoses, a kiddy train, contruction truck rides, tiny tot roller coasters etc.
Hope this gives you some good ideas.