I have relatives in Mpls, and have lived in Chicago, so I've driven at one time or other along the northern part of the river.
Lake Itasca is a modest size state park with the big draw being a stream exiting the lake.
I never tried to stay close to the river as it flowed toward the Twin Cites. But there are lots of lakes and recreation in that part of the state.
At Mpls the river drops over St Anthony Falls (now a dam near downtown), and runs in a gorge from there to Illinois. But south the cities you can't get close to the river until the WI/MN border (where the St Croix joins). From there to I90 there are towns and divided highway on the MN side. On the WI side river access comes and goes. This is probably the most scenic part of the river.
South of I90 I believe there's better river access on the WI/IL side. The Illinois Palisades is a nice scenic area.
My impression is that further south, the river flows mostly across flat land, with a scattering of bluffs. River access and views will come and go.
Starting in Ohio you could go north through Michigan, and approach the headwaters from US2. I consider the northern Great Lakes to be more scenic than the river.
If going south, skirting Chicago, you could see scenic stuff in south central WI before getting to MN. For example, the WI river, Devil's Lake, WI Dells, etc. WI south of I90 is old rolling hills, north of it is flat (like the NS contrast in Ohio).