We dislike interstates and try to do secondary roads. If you have the time that's the way I would go. Here are three options and surprisingly, they are all similar mileages.
Figuring approximately 300 miles/day or 6 hours per day, it would take you 5-6 days of travel but I'd allow more time so you can stop a day or two at various points to just relax. Driving daily wears us out. You can also make variations to these routes.
Also keep in mind that if you're leaving now or later, most public parks will be closed for the season. There should be no problems with RV parks until November. Also keep a watch on the weather.
Taking the interstate directly would put you through some big cities - not fun.
1620 miles - Phoenix/Mesa - take Hwy 60 to Globe; Show Low; Springerville, AZ to New Mexico and I-15. Then cross the highway and pick up 380 east at Socorro, NM to 54 at Carrizozo,NM. Take 54 to I-72 at Griggsville, IL then to I-55 north to Peoria, IL.
1690 miles - Phoenix - take I-10 to Las Cruces, NM. Then US70/82 to Alamogordo, NM and 54 to I-72 at Griggsville, IL to I-55 north to Peoria, IL
1680 miles - via Phoenix and I-17 north to Flagstaff and east on I-40 to I-44 to St. Louis, MO then I-55 north to Peoria, IL