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From there, you can drive through Reno to Las Vegas. This is a long and desolate drive. Don't worry about buying gas on the way, there are plenty of stations nowadays. (Some places try to scare tourists into filling up at ridiculously-overpriced stations because they might "run out" on the way. You won't.)
Vegas. 'Nuff said.
From Vegas, go see the Grand Canyon. You can see Lake Mead if you want, personally I never bothered. If you want, finish up in Phoenix, or drive over to Los Angeles and hit the art museums. (If you go to L.A., you'll probably want to go there before the Grand Canyon, and hit the Grand Canyon on your way back.)
If you can leave your car in Phoenix or L.A., go ahead. From either one, you can hop a flight to New Orleans to go eat something there, and then hop another flight to either Sioux City or Chicago if you need to return to the same city for your return trip to Britain.
If you need to bring the car back, well, personally, I don't think there's a lot along the way, and it will take you 2-3 days of solid driving. Seriously consider paying the car rental place a penalty; at today's gas prices, it will almost certainly be cheaper than the gas.
