I’ve decided I’d like to drive along it for a while and see what there is to see.
With a Madison starting point, if I head straight west I’ll hit the River at Prairie du Chien. But I could head northwest or southwest to hit it at some other point. An overnight stay somewhere is on the table too.
PDC is a fine starting point. Not sure how far you want to drive, or in which direction. You could drive in a loop, taking the highway up to Hudson and then coming down the river on WI35, or vice versa. Or an out-and-back, to PDC, then north, then back to PDC. Lots of options, and again, I’m pretty sure you could drive as far as you want to along the river and enjoy scenic views.
Look for detours along the way, like Granddad Bluff in La Crosse; great views of the entire river valley for miles in either direction.
I’ve been on bits and pieces of WI35, but I don’t know anything about the roads on the Minnesota side. I’d suggest consulting an atlas to see which roads are marked as scenic.
When I head south I am on the MN/IA side until Lansing, then drive on the WI side (until I head inland).
Heading North of La Crosse the WI side may be better until Winona (definitely better bicycling, with the Great River trail on the WI side and I90 on the MN side (at least for a while)
The National Eagle Center is in Wabasha.
On the WI side you have Wyalusing, Perrot, and Nelson Dewey (and to lesser extent Merrick) State Parks. In MN you have Great River Bluffs for nice bluff side view of the river (also Pike’s Peak in IA)