Go to Google maps, click on the 3 lines next to the city search box. Click on bicycling. All paths, trails and shared streets will appear.
What bike are you using? If cross, or MTB is available, then take gravel roads. More scenic, safer (In my opinion).
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You are not going to find a rail trail that goes from Chicago to LA, but there are routes with a couple of days of riding in them.
The Greater Allegheny Passage goes from Pittsburgh to Cumberland (Maryland), and connects with the Chesapeake-Ohio Tow Path which goes on to Washington DC.
The Katy goes the width of Missouri
The Erie Canal Trail goes from Buffalo to Albany. It is fragmented and sometimes you are routed on low traffic roads.
Many states have bike tourism routes. I have done the Willamette Valley in Oregon and Route G in Pennsylvania.
As was previously mentioned, the Google Maps bicycling layer is a great tool. I think Rails to Trails has maps and references. Most states have bicycle touring web sites. You should have no trouble coming up with routes with minimal car interaction.
It looks like you can start in Minneapolis and end up in Carver, Victoria, Lester Praire, Cedar Mills, or Hanover. I think you got options in your own neighborhood.