I did a similar roadtrip ten years ago where I started in Arizona, drove along the southern states to Mississippi and then cut up through Tennessee and Kentucky (stopping at Graceland) to Washington DC and then over to the Great Lakes and along the northern border to Mount Rushmore and then back home through Wyoming with stops at Yellowstone and other spots along the way. I generally avoided interstates (except for a few long boring stretches that I wanted to get through quickly) and tried to eat at local places rather than fast food or national chains.
It took about 5 weeks with obviously a fair amount of time spent in the car. However, it is a good way to get a sense of the whole big country (as well as specific regions) and long drives are not too bad if you are traveling through interesting country that you have not seen before.
Recommendations for such a trip can cover a lot of thing, so it is hard to give you specifics (if want some recommendations on specific places in the southwest, I can help you there). However, some over all basics based on my experience:
Most importantly, I would highly recommend that you buy the book "Road Trip USA" by Jaime Jenson. It is exactly the sort of advice you are looking for - local diners, attractions and recommendations for long distance trips on the older 2-lane highways. It was a great resource for me on my trip (see link below).
Secondly, I have found that the Moon Publications series of state-specific guides books give a lot nice little 'backroad' attractions and points of interests (their Arizona guide includes places that most Arizona residents don't know about). You probably don't want to buy one for every state you will visit, but you may want to consider getting one for those states that you plan to spend the most time exploring.
It is not a bad idea to carry a small tent and sleeping bag so you have the freedom to camp in a campground or just even just pull of the side of a forest service road - this can save money and let you stay in some pretty places. Even if you stay in cheap hotels, it adds up.
Pick out a smaller number of 'must-see' items that are at the top of your list (and especially those that require advanced reservations) and set your dates and reservations for those to make sure you can get in, but don't try to plan every road and every night of your whole trip. One of the things that I enjoyed best about my road trip was the freedom of the open road and just going down whatever route struck me as interesting and staying where ever I happened to be at the end of the day. Part of the enjoyment of being on the open road is the freedom of not having to constantly keep a fixed schedule.
Lastly, I would recommend going in September - weather will still be nice in most places, but you will be past the peak summer crowds.
Have fun and bring lots of tunes