Since there are 5 drivers, you can make some good time. Bathing is mandatory!!
I lived in Napa for several years and I have done the cross country many times.
I've done it in two moths and I've done it in 14 days. 3.5 weeks is tight, what is the rush?
Add in Nashville for Graceland and the Martin Luther King Civil Rights Museum.
Go to Dallas and view the grassy knoll where JFK was shot. In Abeline TX, (I think) there is this Steak House where if you eat a 72 oz steak in one hour (with drink, salad etc) you get it free. It was in the Simpsons. Time clock and stage.
Add in Vegas. (arrive Mon -Thurs and try to snag a 50 dollar room).
See what you can do about seeing the National Parks in Southern Utah. Bryce, Monument, Arches. They are cool.
You got to hit LA. Visit Westwood, Santa Monica Beach, Hollywood.
Past SF, you should drive through the Redwood National Park and see the big Redwoods.
In Washington, go to the Boeing Factory in Everett.
Stop and see Devils Tower from the OLD movie Close Encounters.
When you are going to Rushmore (Free Entrance BTW, parking will cost you 8 bucks? I think you have a free alternate parking site.) Stop by and see the Crazy Horse Monument which is about 15 miles away. There is a nice Former KOA in the area near both.
Also in the Area are Little America where you get a 35 cent Ice Cream (Both ends of Wyoming I think), the Corn Palace in Mitchell, SD for a funky one of a kind deal, and Wall Drug (All testaments to American Entrepreneurship).
In Chicago, see the Museum of Transportation and Industry. THEN go to Ed Debevnics for dinner and watch the wait staff dance and sing on the counter.
Costs:
Lodging:
I just booked hotels for a one month trip NY, Boston, DC and PA and I got incredible deals on the net with and average cost of about 120 a day. this ranges from Westin 4 stars in Boston and DC for about 130 plus tax for 2 adults and three kids to 65 plus tax for a 4 star Westin in PA.
Bring a laptop and when you see wireless hotspot try booking a hotel using Expedia, Hotwire or Priceline. With Priceline, you can get some nice hotels for 50 bucks, but since there are 5 of you be careful because they only guarantee double occupancy. To determine what hotel they are putting you you in when you use Priceline or Hotwire, use
Priceline and Hotwire Forum. Put yourself on Hotwire and Expedia watch lists NOW so they can send you last minute deals. 50 bucks for a nice campsite or 50 bucks for a really nice hotel, I would take the hotel.
National and State Park campsites vary. Get a Let's GO USA. It is written by Harvard Students. It is well researched and will cater to your age group. It even lists itineraries and campsites.
I would budget 100 bucks a night for lodging/camping just to be safe.
When you stop at rest stops, pick up a coupon book for the hotels in the next state and see if there are any deals.
Fuel is pretty easy. 6k miles divided by 17MPG times 4 bucks a gallon. Get a Costco Card / AMEX to gas up or learn about the Flying Js. Fill when it is cheap.
Fuel, Tolls and Parking should be about 1500 bucks
Food. Let's Go will give you some good suggestions. If you are really on a budget, just remember to eat at restaurants for a reason. In Tennessee I stopped by this stop that had a diner selling Po Boys and Hoagies. Across the building was a national chain sandwich. I took pictures of the hoagies and I can even remember the name of the chain.
Don't waste too much on fast food. It adds up. Believe me, where I live, a kids happy meal is 6.50 and a meal deal IS OVER 10 BUCKS. When my kids want fries, it ends up costing me thirty bucks. For 5 people, a Costco pizza for 10 bucks can be a life saver. Hot dogs and soda for 1.50 also make an economical snack. Do bring a camp stove and treat yourself to some nice steaks/meals.
In The California Redwood National Park area there is this logging town (Free Tour on weekdays). The general store has incredible meat at incredibly low prices (6 bucks a pound last time I was there, but probably higher now). Get yourself a 2 inch thick T Bone. Now I have been to Vic Stuarts, Mortons and Ruths Chris and paid a hundred buck for a Porterhouse. This was almost as good.
North of SF there is Johnson's Oyster Farm where you can eat fresh RAW oysters out in the parking lot or bring them back to your campsite. Bring beer!!!
While you can certainly do food for 10 a day, you can also do 100 a day per person. I would budget 20 per person as being pretty comfortable. The key is getting 5 people to agree on how to budget. I took a trip up the Alaska Highway with 7 people on a school bus (4 10 year old nephews on my dime for 1.5 months). When we got to Anchorage, we met up with a friend who flew up from SF. He immediately wanted to go to a restaurant. Nothing on the menu for less than 10 bucks. Nice that he paid for himself, but try that times 7 people, three times a day and for 45 days. Set rules (that can be flexible). That can save you hundreds, really get the feel for the local fare and lead to less aggravation. Also, prepare and have good low cost and interesting alternatives that you can really enjoy and remember.
Good travels.