Welcome to the Twin Cities!!
It might be a little bit of a drive to some locations, but Three Rivers Park District (http://www.threeriversparkdistrict.org/) has some really great programs, including some for your daughter's age, and all of their nature centers (they have indoor buildings with exhibits and play areas) are free to visit and explore.
And don't forget about library storytimes...Central in downtown Mpls has tons of special events too!
Other ideas that won't break the bank:
Toddler Tuesday at Mall of America
Science Museum Preschool Playdate Tuesdays (St. Paul)
Macy's 8th Floor Holiday display (Nicollet Mall) free, The Nutcracker is the theme this year
Family Village (New Hope) free play area and great classes
Sovereign Grounds (South Mpls) coffee shop with play area
Baby Loves Disco (Mpls, Foundation Nightclub) Dec. 15
Please also check out http://twincities.parentzone.com (I'm a local freelance writer...and SAHM to 2 and 3 1/2 year old girls...and that's my current project). There's a lengthy list of Classes for Kids. Within the Family Fun area there are subcategories for indoor play areas, free things to do, and there's also a calendar of local events. You'll also find a list for Playgroups and Parent Groups and you're likely to find one nearby. Good luck!!