E.T.
We live south of Minneapolis/St. Paul (aka The Cities) and go up there quite often.
I would suggest staying in the Bloomington area near the Mall of America. You can get to pretty much anywhere in the Twin Cities area within 20-30 minutes. There is a light rail station at the Mall of America. You can park free there and take the light rail to Target Field for the Twins game. Although downtown Minneapolis is pretty cool, I don't think you would spend a lot of time down there other than the Twins game unless you were going to go to a theater or the art museum. A lot of the things I'm going to suggest are in St. Paul or the suburbs.
If you like Thai food, Ruam Mit Thai in St. Paul (not far from the science museum) is the best! It kind of looks like a hole in the wall, but it is consistenly one of the highest rated restaurants. Mikey's Diner in St. Paul (not far from the same area) is an icon. It is a very small greasy spoon restaurant in an old train car. It feels like something out of a 70s sitcom. Midtown Global Market in Minneapolis has a huge variety of ethnic foods and shops.
We love going to the Minnesota Zoo (in Apple Valley) and the Como Park Zoo (in St. Paul). The Minnesota zoo has a summer concert series that pulls in some big names.
The Mall of America (in Bloomington) is probably my least favorite place to go, but teens love it! They have some of the typical chain restaurants like Rainforest Cafe and Bubba Gump Shrimp.
If you can get tickets for a St. Paul Saints baseball game, definitely do that. They are a minor league team and their games are lots of fun.
The Waterpark of America (in Bloomington) is pretty good, but I thought pretty pricey. If you are going to the Dells, you can probably skip that.
Grand Avenue in St. Paul is a fun area to walk around. There are some good restaurants and a yummy ice cream shop. Stillwater is also fun. The Lake Harriet and Lake Calhoun areas (in Minneapolis, but not too close to downtown) are good for outdoor activities like canoeing. There are often outdoor concerts in the Lake Calhoun area.
If your family likes to read, definitely check out some of the independent book stores in both Minneapolis and St. Paul. If you like theater, there are some great theaters to go to in both cities. I know there are going to be several off-Broadway shows there this summer (Lion King is one I know for sure.)
The science museum in St. Paul is really good. The Walker Art Museum and sculpture garden in Minneapolis is nice.
I've never done it, but some friends of mine have several times gone on the Wabasha Cave tours in St. Paul. There are some caves under the city that were used by gangsters during prohibition.
Valleyfair Amusement Park (in Shakopee) is a lot of fun for teens. It isn't huge, so you can easily do it in a day.
When you drive from the Cities to the Dells, drive along the Mississippi River down to LaCrosse, Wisconsin. It is a really pretty drive. You can see some of the locks and dams on the river. There is an really nice National Eagle Center in Wabasha.
Take a look at the Explore Minnesota website. There are tons of things to do!! Be sure to check out a calendar of events for the time that you will be there. There seems to be little festivals of all kinds going on all summer long. Have a great time!