Driving straight thru to the west coast you are looking at about 3,000 miles one way. Assuming 500 miles per day (a long day on the road by the way) that is 6 days travel each way, or 12 days total. Nearly half your allotted time will be on the road, and that does not count all the side trips to see the various attractions. 3 weeks would leave you very little time for sight seeing or staying at one place more than a day or two. 4 weeks would be better but still on the move a lot. Not trying to discourage it, just make sure you realize what you are in for.
There is a lot to see across NM, AZ, CA, OR, UT, NV, WY, ID, MT, and CO. Just have to look at all the options and make decisions on everyone's priorities. You can't see it all.