Hi there Davenport!
Hey, that's my home town. Grew up there. Still have relatives there and get back on occassion. Haven't driven to Disney from Davenport in quite a few years, but the Interstate systems are still there and it's not a bad drive, but it is a long drive. We drive it all the time now from Indiana, so a lot of the route I'm showing is still the same as from Davenport, and very familiar to us.
Using my mapping software it shows you're 1237 miles from the I-74 bridge at Moline to Disney's FW. The recommended route is as follows:
I-74 to Peoria, (be sure to take the I-475 by-pass. Don't go through town. You'll get slower speed limits, more curves, and more traffic as you get close to the river if you go that way. The By-pass is only one mile longer than the route through town, but it's well worth it)
Just pass Peoria, catch I-155 south and follow it to the junction with I-55
Take I-55 to junction with I-64 southeast of St. Louis. You can either take I-55 to I-255 to I-64, or save a little bit of time and take State Route 4 off of I-55, exit 33, due south 15 miles to it's junction with I-64. I haven't driven that in a long time so don't recall the actual road there, so you may want to stay on interstate rather than get on a state route.
Follow I-64 to I-57
Follow I-57 to I-24
Take I-24 all the way to it's junction with I-75, southeast of Chattanooga.
Take I-75 into Florida. If you hit Atlanta anywhere near rush-hour, take the by-pass. Otherwise it's OK to go through town which is shorter and won't be a big problem. But by all means, don't do it if it's rush hour. When we go down, we always plan our trip timing to avoid Atlanta during rush hour.
Be sure to take the I-475 loop around Macon. You don't need to go into town. This is the way most traffic will go anyway. There's still construction going on there now, but it may be done by the time of your trip.
In Florida, south of Ocala, you'll finally hit the Florida turnpike toward Disney. Follow it East. The tolls aren't bad.
From the turnpike's junction with I-4, go west toward Tampa (don't go toward Orlando). You'll actually be going south at this point Watch that exit from the turnpike to I-4, it's a sharp turn. (Note, instead of going that route, you can also exit earlier and take route 27 south to junction with 192, then go East to the Disney Magic Kingdom entrance, and there to FW. Hwy 27 is pretty good, but traffic is getting heavier. Be careful as there are a couple of stop lights that you can come up on pretty quick. The construction on 192 is now all finished, and it's all 4-lanes now. Traffic is not near as bad as it use to be. It won't save you any time getting to FW, but is shorter mileage, and keeps you off of that I-4 traffic which can get pretty busy, especially at rush hour.)
If you do go to I-4, then follow it about 5 miles to Disney's Epcot exit. (It's hwy 536)
Head into Disney and follow the signs toward the Magic Kingdom and toward Ft. Wilderness.
You won't hit any really bad mountains going this route. There'll be some stuff, especially around Chattanooga but nothing really bad. I don't think these are as steep as some of the stuff we routinely hit on I-75 when we take that via the Knoxville route.
We just got back from an extended RV trip (10 weeks) which included 11-nights at FW and another 30 nights off-site in a nice RV resort. (Encore, Hwy 192 West out toward Hwy 27). We got a special promotional rate: $99.00 for a month. Boy, would love to be able to get that rate on a regular basis. (Regular rate is about $500/month)
The route Breick suggests, I-74 to I-57 and then down is OK too, but it is 20 miles longer. But they are good roads.
Your Streets & Trips software takes you through the Smokies? I presume that means I-74 all the way to I-75 in Cincinnati, and then on toward Knoxville and down that way. That doesn't make sense, that's almost 100 miles (1332 miles total) further going that way. If you did go that way though, you'd go right past our house. Wave as you go by.
Have a great time.