Walker Art Center is cool and has free first Saturday admission. Looks like to celebrate the reopening of the sculpture garden on the 10th they're doing free evening admission the 1st through the 10th. If you like art, the Minneapolis Institute of Art is also free and great (I think ~$5 donation encouraged).
To echo olliecjb, walking or biking around the lakes is awesome. Plenty of great places to get a beer. Get to NE Minneapolis and hit up any of Indeed, Bauhaus, 612 Brew, Sociable Cider, Able Seed House, Northgate, or Insight. All reasonably close to each other. I think they all have bike racks, too. I've walked between the first four in one lovely day. Or go down to Surly and take the tour.
There's a nice walk on the outskirts of downtown Minneapolis including West River Road, the Stone Arch Bridge, Main St. and Water Power Park, and the Hennepin Ave Bridge over Nicollet Island. Makes a little loop across the river and takes you through historic Saint Anthony Main. Then stop off at Gold Medal Park which is nearby for a bit. If it's warm, walk down to Izzy's Ice Cream, which is a bit spendy but it's good and you get the bonus Izzy Scoop. Behold the new stadium (ick), the Guthrie Theater and it's funky cantilever protrusion, and get a good view of the riverfront. Mill City Museum is pretty cool and is also right in the area ($12 admission). Get to walk through an exploded flour mill and get an excellent view. This area pretty well embodies how I think the city feels.
Closer to St. Paul, but the Como Park Conservatory is a great place. Free ($3 suggested donation) and you get to walk through a bunch of climate controlled areas where some fairly exotic plants are growing. I enjoy it more in the winter, because it's warm and humid, but worth the trip any time. I'm pretty sure their Japanese garden is open now but isn't in winter. Como Park and Lake Como are right there and are really nice areas, too.