Liz Weddle's recommendations are the Mia or the Walker Art Center. The Mia is free admission with art from a variety of time periods and cultures. The Walker is contemporary art, and admission is $18, but it also has a free sculpture garden which is where you can see the classic "Spoonbridge and Cherry" sculpture.
Greg Rosauer recommends you visit the Cathedral of Saint Paul. One of the finest examples of Beaux Arts architecture, the Cathedral was designed by E. L. Masqueray. Guided tours and self-guided tours of the building are available.
Becky Halberg's favorite local attraction to visit with her family is Como Park Zoo & Conservatory. There is a little bit of something for everyone—lions, tigers, and bears (oh my!)—as well as greenhouse tropics and beautiful gardens to walk through. The best part—it's free admission.