I think you can see a lot in three weeks. If you are willing and able to do a few mini-trips that make up the three weeks, you could get a pretty good view of the immense diversity in the county. Since you are starting in the NE, I would visit NYC, DC, PHL and BOS, all can be done via train or bus, no car needed.

You really should see the Rockies. Go to DEN and visit for a couple of days and then drive to SLC. The scenery change is amazing and Glenwood Canyon is one of the most beautiful pieces of road in the county. Lots of places to stop and photograph.

Short flight to SEA, see the sights there and rent a car to drive south. I wouldn't spend too much time in LA. I've been countless times and am generally pretty bored there. SF, PDX and SEA all have a lot more beauty to them. As mentioned, the Columbia Gorge is fantastic and I can't help but promote PDX to visitors, it's a nice place. Drive out to the coast of OR and then head south. Not that LA is all bad, there are some cool things to say, but I'd be bored out of my mind with 3+ days there.