I did exactly what you are thinking of doing last year.
We bought a car in LA and did a loop like this
LA-San Fran-Tahoe-Vegas-Grand Canyon-Santa Fe-Colorado-Utah-Montana-
Calgary-Vancouver-Seattle and down west coast back to LA
That was 15000 miles all in (thats what the clock reg'd)
Be very careful about thinking you can drive from east coast to alaska. You can but you had better enjoy driving!
For $50 you can get a parks pass, which allows access to all the national parks. Its definitely worth the money, if you are into that sort of thing. 3 months is not that long to travel around the states. we had great plans to drive all over the place but you just cant do it, its massive.
My one definitive thing to do would be to drive down highway 1 from washington state to san fran taking in the redwoods on the way.
Enjoy yourself!