There's more in the USA than three weeks worth. Or five for that matter - I just finished a 24 state epic in five weeks and barely scratched the surface. My advice is not to try to see everything. You can't do it and you'll wear yourself to a frazzle trying.
Las Vegas. Good choice. It's way over the top corny, but it's great fun, so long as you don't blow a bunch gambling. I got a room at the Las Vegas Hilton (yeah, just like in the song) for $52 a night. On the way back I splurged out and for $75 got a room at the Paris, which is right on the Strip and tres jolie. Expedia is good for this - I enter my criteria and then rank them according to guest rating, which sends the dogs to the bottom, despite whatever tricks the hotels themselves use to inflate their appeal.
You could use Vegas as a base - there are tours to nearby attractions such as Hoover Dam and Grand Canyon, but you definitely need a few days to get the full impact, take in some shows, just wander up and down the strip. You could hire a car and do a tour of your own. I'd suggest checking out of your hotel, driving via Hoover to Grand Canyon village, staying the night, and then drive back via Flagstaff and Route 66. Two easy days, lots of great scenery. Book accommodation at the GC first - it can get tough to find. Or in Flagstaff, but the logistics makes that a bit trickier.
New York. I could spend a happy week there. A lot of the good stuff is free. Walk over Brooklyn Bridge, take the Staten Island Ferry, wander through Central Park, visit Times Square in the evening. A thousand great ethnic restaurants. The subway is cheap and convenient. Have lunch at the Carnegie Deli, browse through the Strand Bookstore, have a steak at Mickey Mantles on Central Park West and they'll give you a 20 oz souvenir beer glass. I stayed at the Waldorf there, believe it or not, after using a sale code I got here or on Flyertalk.
Lots of other places. Chicago is an architectural delight. They do a boat cruise up and down the river past some fantastic skyscrapers which is wonderful - even if you don't like architecture. Oak Park has some of Frank Lloyd Wright's masterpieces. The parks and museums in the city centre are great. The Bean in Millenium Park is a great photoshoot. Have a Sunday Brunch at the Signature Room atop the John Hancock. The view is mind-blowing. Or just go up for a drink - the elevator ride is free, but they charge steeply for a beer. Macys on State Street - the old Marshall Field store - has a Tiffany dome that is just awesome and the chicken pot pie in the Walnut room is a delicious tradition. There's also a Levenger store there, and if you like gadgets and journals and books, it's a must.
Washington DC for the iconic sights and the Smithsonians. If you are a plane nut like me, the Air and Space Museum and its enormous annex at Dulles is a religious experience. There are various bus tours to give you an overview - I recommend the twilight tour leaving from Union Station
You could do Fort Worth for a taste of Texas. There's a Hyatt right in the Stockyards which is a good base for the corney tourist show and longhorn mouster. The Kimbell Art Museum is a gem, and I adore the Japanese Gardens there.
Or San Francisco. God, but I love that city. There's the tourist things like the Golden Gate Bridge - and the day I rode a bike across the bridge with a beautiful blonde is still the best day of my life - and the cable cars and Alcatraz and Fishermans Wharf and chinatown. Or just walk around, enjoy the funky atmosphere, those crazey hill streets. I love it.
The straights of San Francisco by
skyring, on Flickr
See if you can take in a ball game while you are there. The Yanks make it an experience, even if you don't know the rules or the players it's still a chunk of Americana that is legendary.
I'd suggest (say) four nights in five of the above cities, taking three days for each city and a day for travel. That's at least a taster.
