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A trip like this is close to 1000 miles, which really means it needs to be measured in days not hours. It will be 2 pretty full days on the road.

Your fastest route looks to be I-20 to Atlanta, then I-24/I-57/I-64 through Nashville to St. Louis, then straight north on US-61/US-218 to Iowa City. The last section is not interstate, but is 4 lanes for the vast majority of the route (also nicknamed the avenue of the Saints, as it connects St. Louis and St. Paul) and should work quite well.

Paducah Kentucky would likely be about your halfway point and would make a good stopping point, but you could get as far as Mt. Vernon, IL if you get an early start on day one and want short second day. If you get a late start on day one, you could still make it to Iowa City on day 2 as long as you get as far as Nashville.

Weekend vs weekday doesn't make a big difference. Weekdays may see slightly cheaper hotels, but you're less likely to see traffic problems on a weekend.