Thomas, 

I would take US 17 down to Williamson, US 64 over to Raleigh, I-40 from Raleigh all the way to Nashville where you would catch the Briley Parkway on the East side of Nashville. Take the Briley all the way around the city to I-24 and take 24 up into Illinois. In Illinois take 24  to I-57 and then 57 up to Mt. Vernon where you catch I-64 to St. Louis. To get around St. Louis, we usually take I-255/270 around to the South. From St. Louis, take I- 64 and I-70 to Kansas City, where you would head NW on I-29.  29 will take you up to Souix Falls where you would catch I-90, which will take you to Spokane.

You'll miss most of the Nashville traffic by taking the Briley PKW around the city, The traffic in the St. Louis area can be pretty bad, so I'd consider the route around to the South. I don't think you'll have too much traffic problems with traffic in Kansas City or Sioux Falls except during the rush hours. 

Hope this helps,

Jim

I almost forgot, depending on the time of day you get there, you may be better off taking I-640 around the North side of Knoxville, rather than going through on I-40.

Weather-
Pay attention to weather reports as you drive. We got into a storm that got progressively worse and heard tornado warnings so we pulled off at a truck stop to wait it out. A tornado went across the highway about a mile back in the direction we had just came from.

Fuel-
I get fuel at large stations or truck stops or if needed in a small town will drop the 5th wheel at a church, school, store with a large parking lot and drive to a small station to fuel up. If we've unhitched at a campground/RV Park I will fill up in town (if close by) before hitching back up.

Sights along the Way-
Give yourself plenty of time if possible to see the tourist sights along the way... you never know when you might get back that way. Unfortunately we didn't allow enough time to tour Washington DC so we ended up driving around for a few hours and only got to see the outsides of the buildings...."there's the White House...where ?... Over There ! ...Oops too late...". We had a schedule to see relatives in different cities. We learned that 4 to 5 hours is plenty of time to catch up with relatives. More than a day can sometimes be way too long to spend with them (can you say, Bored).  I would have rather spent the days in DC touring the sights.  It's been 19 years and we haven't been able to get back to DC.  It's on our bucket list.

Bear II, I don't think he'll have time to see the sights along the way as it's about 2,800 miles out there and he only has 7 days to make the trip. He'll be logging 8 to 10 hours of driving each day the way it is, and that's only if he doesn't have any delays.

It would take us about 3 weeks to get out there, the way we travel.  

Jim

Thank you all for this valuable information.

We start the first leg of this adventure tomorrow heading to NC.
On July 5th we head West. I'll try to post our progress as we move along.

Our trip is now history. Both the Ram pickup and Bighorn fifth wheel handled flawlessly with no mechanical issues. We packed a lot in for this first RV experience which we looked at as "Lets see what this RVing is all about" 
Result is we loved it. From this point on we will do shorter trips with more layover times. A brief description follows. Thank you all again for the tips, every one of them was put into action.
Tommaso

We hitch up the Bighorn RV and drive to the Outer Banks, NC for a week of bonding with Katie's family, June 28-July 5th

July 5th we start to drive to Spokane WA. for a family wedding, Katy & Ben, arriving Jul 11th.

Jul 12th wedding day, was the catalyst that drove us here in the first place.Thank you Katy and Ben for providing us this destination.

Jul 13th quick spin to Aloha, OR for a two day visit with friends Ray & Lori Hennings.

Jul 16 depart for Denver, CO to visit Bob & Terene Stewart for 1 1/2 days.

Jul 20 leave for Madison, OH for a one day visit with Ron & Helen, arriving July 23rd.

Jul. 24th leave OH and arrive Home Sweet Home, Liverpool, NY 8:30PM

Yahoo, 7,046 miles---29 days---21 states ---NY, PA, MD, VA, NC, TN, KY, IL, MO, IA, SD, WY, MT, ID, WA, OR, UT, CO, KS, IN, OH---740 gal's diesel fuel = $2996.42 worth.

Total trip value was priceless, being entertained by life on this planet.

I agree with Bill.  What a first adventure!  And you're ready for more... I like that attitude.

Now here's the number cruncher in me (always looking for new data to tweak the budget forecasts)....

9.52 MPG average... $4.05/Gal.  OK, you clearly did a lot of towing vs running about sans trailer, so a more leisurely 1000 miles per month would be looking at $425 per month on the high side for an budget estimate for fuel.  While everybody has different driving styles, and different ratios for towing vs bobtailing... Your numbers reasonably confirm my targets for what we should be setting aside for fuel. Thanks for providing the raw data for me to work with.

Here's to your next adventure... may it be as memorable as the first

Brian