I think you need to start with your timetable, prioritize what you want to see, and then go from there. You don't have time to absolutely blanket the USA!
Assuming you arrive in San Francisco delightfully refreshed on July 20 (not
around July 20th), you have
sixteen days to make it to Disney World in Central Florida. The shortest route -- which is probably NOT where you would want to go -- is a little less than 3,000 miles. Driving hell-bent, with a decent pain threshold, that's 5 long days of driving.
The shortest route is the southern route, basically along I-10 through the southern tier of states. With a few detours, that route could take you to Vegas, Grand Canyon (and the prettier areas nearby like Sedona and Monument Valley -- we just returned from two full weeks there), across Texas, through New Orleans, and on to Florida. There is no shortage of places to spend extra days along that route -- you could easily spend weeks exploring northern Arizona and southern Utah alone.
Another route would be more northern, and almost 4,000 miles -- SFO across northern CA and NV to Jackson Hole/Grand Teton/Yellowstone, to Mt. Rushmore, across the upper Great Plains to Chicago, then down the Mississippi River system to New Orleans and east to Mickey.
And then there is a combo route: SFO to Vegas, Grand Canyon and Utah, Yellowstone etc, Chicago, New Orleans, Mickey. That's about 4,300 miles. That would give you a very representative look at parts of the US that you have not visited. It's a long grind, but would be worth it.
On that route, first of all I would spend at least two full days in San Francisco itself. GREAT city, one of my all time faves.
From SFO, I would drive 4-5 hours and spend another 2 days exploring Yosemite National Park in east central CA. It's the prettiest place in America, but it will be a tourist madhouse in July.
You'd want to spend a couple of days in Vegas, at least 2-3 days in Jackson/Grand Teton/Yellowstone, and probably at least one full day in the Rapid City/Mt. Rushmore area. The best way to get from Yosemite to Vegas is up into the alpine area via Tioga Pass (north of the valley, still inside the park) and then down the east side of the Sierras to Vegas.
I would also spend a couple of days in Chicago, and a couple in New Orleans.
A couple of things about that combo route.
One, it's heavily oriented toward our biggest National Parks, so you would want to purchase an
America The Beautiful pass. It costs $80 currently (might be going up October 1, 2015...not sure), but it will get your entire carload in any federally-managed recreational land (including all National Park Service sites) for a
full year from the month of purchase. You can buy one at any national park. Some of these parks (Grand Canyon, Yellowstone, Yosemite) are increasing their entrance fees to $40-$50 per car on October 1, so that pass will save you big bucks.
Two, if you visit those parks, you will want to stay INSIDE the park. Outside the park is WAY outside where you want to be.
National park lodging can be booked one year and a day from arrival, and many of the parks sell out the first day, so you will want to plan carefully and book as soon as you possibly can. In Yellowstone/Teton, you could cobble something together between the two if you couldn't get exactly what you want. One night in Teton or Jackson, WY and two in Yellowstone would work beautifully.
Yosemite will be the toughest, and you should try for Yosemite Lodge -- moderate price, good accommodations. Don't take any of the outlying areas (Wawona, etc) At Grand Canyon, any lodge except the El Tovar Hotel (expensive) will be fine. In Yellowstone, the Old Faithful area, Canyon, or Lake Yellowstone area are all conveniently located. Mammoth Hot Springs area is well north and not that convenient. Anything in Teton or Jackson, WY is fine for Teton. No lodges at Mt. Rushmore, anything around Rapid City, SD is fine.
For National Parks info, go to
www.nps.gov . Specific websites are as follows:
Yosemite -
www.nps.gov/yose
Grand Canyon -
www.nps.gov/grca
(there are many other NPS sites in N. AZ and So. UT, between Grand Canyon and Yellowstone/Teton)
Grand Teton -
www.nps.gov/grte
Yellowstone -
www.nps.gov/yell