Do the math beforehand to get a good estimate.
My family has made this trip several times, and it’s not cheap.
That said, hotel and food is really where you can save. Gas isn’t going to make much difference. In other words, you can save thousands in the food/hotel area, but with gas you will save $1-200 max.
For example, on our last trip we drove ~7,500 miles over a 3 week period. We ate restaurant food usually once per day, and we stayed in reasonable hotels.
7,500 miles / 25mpg = 300gal; 300gal * $3 = $900.
Food for 4: $100/day * 21 days = $2100.
Decent hotel for 4: $100/day * 21 days = $2100
Car rental from Avis for a month: $900
Of course, there were nights when we spent $60 at the Days Inn in the middle of nowhere, and there were nights when we spent $130 in a downtown Holiday Inn.
My favorite was the Hilton in Vancouver, Wa — it was brand new and the demand they were hoping for never materialized, so they had excellent rooms for $120.
Food probably didn’t cost $2100, as we ate more junk food than restaurant food, but it was important to understand how that one builds up.
This is a realistic estimate for a fairly lazy family who didn’t want to live in the car, didn’t want to deal with bedbugs, and was too lazy to cook. Clearly if we had been poor college students doing the same thing, we would have bought some basic staples for food and lived out of the car and saved ~3-4K.
Do yourself a favor and splurge for the extra gas and rent a comfortable vehicle. Your family will thank you.
And I did say Rent. I never drive my own vehicle on one of these expeditions. Who wants to have to deal with a breakdown somewhere on the Oklahoma Turnpike in the middle of August?
ETA: Take your laptop. All cheaper hotels have free wifi, and the better hotels give it you for a few bucks a night. It’s awesome to be able to sit in the comfort of your hotel room and plan the next day and then shop around online for hotels in the next destination. Very nice. It allows you to change your plans on a whim.