Average ice out off the lakes in the Twin Cities area is around April 15th. It can vary from late March to late April. Trees do not leaf out until about May 1st. The temperatures in early April have at times topped 90 degrees but last snow fall can be in May. It can be iffy but generally May is pretty nice but not swimsuit weather.
What to do in Minnesota? You can Google any suggestion for details.
Minneapolis/St. Paul is loaded with museums and cultural attractions if urban exploring is your thing. They also have great lakes and parks and numerous walking/biking trails -- more than you can imagine.
Mall of America with Ikea nearby. This is a shopaholic's heaven and a unique attraction. Just to walk it is at least a 3 mile hike. I have to mention it because I was involved deeply in the mall's design and spent three years of my life working on it.
Duluth is the other city worth time and exploring and of course is the springboard to driving up the north shore. If you are a birder, Hawk Ridge Nature Reserve is the place for spotting.
Starting in the southwest you can explore Pipestone National Monument and Blue Mounds State Park (for camping and viewing the bison herd).
A drive from La Crosse, WI up Highway 61 along the Mississippi to the Twin Cities is scenic and historic. You can stop in Wabasha, MN and visit the National Eagle Center, a very impressive place to learn about and see our national bird. Lunch at Slippery's in Wabasha and have a Walleye sandwich. Slippery's was the place used in the movie Grumpy Old Men. Or if you want a touch of Scandinavian Minnesota the Anderson House will do. The City of Red Wing on the way is one of the 25 best historical destinations according to National Geographic Traveler magazine.
You can go to Itasca State Park where the Mississippi River begins and say you actually walked across the Mississippi (because you can there).
You can visit open pit iron mines on the Iron Range and actually go down a mine at Tower Soudan State Park.
On to Ely, the stepping off point to the Boundary Waters Canoe Area. If that interests you, you can see it from the road or you can make arrangements to go into the BWCA via canoe and experience nature at its most primitive. The Dorothy (Root Beer Lady of the BWCA Molter Museum is interesting. The Bear Center is nearby.
The drive from Duluth to the Canadian border is the best. There are several parks and scenic water falls along the way right up to the border. You can drive the 60 mile Gunflint Trail from Grand Marais, MN and again touch the BWCA.
On to Wisconsin.
The Apostle Islands on Lake Superior is a pleasant stop and you can rent kayaks there and take a guided tour on the lake.
The National Forests in northern Wisconsin are great for primitive camping. Most campsites are on lakes in conifer forests.
Door County northeast of Green Bay is the Midwest equivalent of visiting Cape Cod/Maine shore type of places.
Milwaukee's Santiago Calatrava designed Art Museum is a must stop in my opinion. There is no other museum like it.
You've got the Wisconsin Dells if you are in to that sort of thing. There are boat tours, duck boat tours of the rock formations. There are also tons of water parks and trinket shopping. It is actually the premier destination for taking kids.
The area southwest of Madison, WI is worth exploring. The area centers around the historical Swiss village of New Glarus. I like to go there for the New Glarus Brewery, one of the best brews in the country.
The bonus. If going to Minnesota and Wisconsin you might as well include the Upper Peninsula of Michigan with Porcupine Mountains Wilderness State Park, Tahquamenon Falls State Park, Pictured Rocks National Monument and Mackinac Island.
Leaving you with a drive up the North Shore of Minnesota...
Gooseberry Falls
Gooseberry Falls
Split Rock Light House
Cross River
Cascade River SP
Devil's Falls, Judge Magney SP
Pigeon River SP on the Canadian border