Welcome aboard the RoadTrip America forum!
For some quick ideas on what's available along your basic route, check out the similar threads at the bottom of this page, and for tips on travelling on a small budget, be sure to read Bob and Gen's Art of the Cheap Road Trip.
I'd count on at least 5 days to get across country, particularly if you're alone. But that would leave almost no time to see anything except through the windshield. If you could possibly add a day or two more, then you can really enjoy some of the amazing sights this country has to offer. What you want to see is really up to you and your particular tastes, but on a full transcontinental trip like this you should be able to get to many places without too much of a detour
AZBuck