You don't have to stick to one side or the other. The shore varies as you go north. I lived in Chicago, and visited relatives in Minneapolis, and often preferred the river north of I90 to taking I94. Even rode that part on a bike trip between the cities.
In Illinois the best views and access to the river are at Mississippi Palisades State Park. Galena is away from the river, but has tourism charm based on historic lead mining.
From there I don't know if trying to stay close to the river is worth the effort or not. SW Wisconsin, even as far east as New Glarus, might more interesting. You can drive down to lower Wisconsin River to its mouth, and resume the Mississippi drive.
WI 35 stays along the river to La Crosse. The west side also looks close, though it might be a lesser quality road.
US61 is a major road on the west side until Red Wing. On the east, 35 is next to the river where it can, but heads inland where necessary. Here its enough of a valley, that grades in/out can be significant.
From Hastings you aren't close to the river until you get to St Paul.
