It's going to be excruciatingly expensive. My sister and BIL had thought about buying an RV, and just wanted to "try it for size", so they rented for a long weekend from Cruise America, drove with the kids from NJ to DC and back.
A few notes--the RV was NOT clean at all. In fact it was downright skeevy. My BIL got them to knock about $100 off the rental because they had to go to a store and buy cleaning supplies to clean it--the bathroom was gross with hairs all over the toilet, etc.
At any rate, their trip in the rental was fairly short in time and distance, and it still cost them more than driving their own car and staying in a very nice hotel would have.
In the long run, they did determine that it was a good way for them to travel, and they bought a "Rolling Hilton"--a gorgeous 40' RV with slideouts and an onboard shower. (I'm envious!!)
Now that they have their own RV, they drive down to FL to see us in it--just kind of meandering along and stopping at various educational sites along the way, battlefields, museums, etc. My sister homeschools, and they research "sidetrips" with content relevant to the current curriculum. (And for you anti-home schoolers, she's got a PhD in psycology and her disertation was on cognitive development in children. She's also an experienced educator, and the children all participate in "extra-curricular activities" with other kids, karate, Scouting, dance, soccer, riding, etc.)
Driving their own RV makes it much more affordable than renting one, even though it only gets about 6-7 mpg.
R/T from their home in CT to my home in FL costs them about $1000 in gas alone. Going cross country would probably be at least couble if not triple that or more in gas, plus the rental costs for you. They charge milage after the first 800 miles, at $.25/mile if I'm not mistaken. That's going to end up being a $10,000 trip if you take three weeks to do it--don't forget the cost of campgrounds which can add up in some areas, and driving/parking in some cities will be almost impossible. And doing it in less than three weeks is not going to be feasable.
Here's my advice. Don't try it. I've driven cross country about a dozen times, it's exhausting, even doing it over an extended period. When you drive cross country there are a LOT of times that you'll spend hours looking at cornfield after cornfield, desert after desert, or just rolling nothing after rolling nothing. You'll get excited to see a billboard or a dead armadillo in the road. You'd be better choosing one geographical area, say California and Nevada, and flying there. Once you are there you can pick up the RV and spend a week to ten days exploring that area. You can keep you milage/gas costs much lower that way, and really have time to see and do things.
In all honesty, you just couldn't do much more than wave as you drove past to see the entire country in three weeks time. You'd need more like six weeks, and if you were going to be gone that long, you'd be better off simply buying an RV.
Anne