Generally speaking I would take a route going to the southwest (NM, AZ) and possibly SW Colorado (Durango to Telluride). This desert area will be pleasant in the late spring. Then head north going through California which in mid May the coast will be clear and warm and up through the state and into Oregon and Washington and then head east into Glacier NP and then south into Yellowstone and the Grand Tetons where in mid summer the wildflowers will be in bloom. Option then of heading south to Salt Lake and Utah desert or heading east to other sights.
Summertime in the high country of CO and NM is OK but Utah and even eastern CA is something else entirely. The CA coast and Oregon and Washington will also provide pleasant weather along the coast and at the higher elevations going inland.
California is one of only 3 states in the USA where 100% of the coastline and all the beaches are open to the public. Great camping along the coast and especially nice in San Simeon and Big Sur, Mendocino, and Orick (Prairie Creek Redwoods SP). Many beaches that are completely wild in the north half of the state and few places on the east coast are like this and where they do exist the camping is usually highly restricted.
I could lay out a great vacation in California by taking one of the state parks maps and drawing out a route. They include high and low desert country, lakes, high mountains, forests, and beaches. No other state provides this kind of diversity.