Hi Jessie,
I think that you would likely want to take longer than 3 months to do all of the routes on the Road Trip USA site. We regularly hear from folks who want to see it all when the come to the US. I don't blame them. With all that I've seen I still want to do more! But if you don't have more time (and the necessary $$) I suggest cutting out a good bit. Yes. you could actually cover all of those routes. But you won't have the time to enjoy it.
For reference, I had done a road trip from Pittsburgh, PA to Seattle, WA. My route was very similar to the Route 8 (The Oregon Trail) on the Road Trip USA site. I took a total of 3 weeks. I was able to spend 2-3 days at Badlands National Park, Yellowstone Natioal Park, Grand Teton NP and Glacier NP. I was also able to spend a day or more at Jewel Cave, Wind Cave, Deadwood, Mt. Rushmore, Crazy Horse, Wounded Knee and more. In between each stop I did heavy driving (8-12 hours a day). In 3 weeks I saw a lot. I had enough time to spend at each stop to feel that I saw everything that one should see on a first visit.
With 3 months, I think that you could make a nice loop around the country. Again, using the Road Trip USA routes as examples, 3 months should allow you to do Route 8, Route 1, Route 10 and Route 6. Of course that all depends upon how many places you plan to stop at along the way and how much time you'll spend there.
For a major natural attraction or a city, I generally find that 2-3 days works well.
Regarding the transporation. Yes. We say that non-residents buying a car is not a good plan. The car rental companies to offer discounts for long rental terms. The rates are less (per day) than if you rented for a few days. I did a search on one of our national car rental companies. For 3 months you can get a compact car for about $1800. That's $20 a day. If you were to buy a car for that price, it could be unreliable. You'd be taking the risk of it breaking down in the middle of nowhere. With a rental, you at least know that you will have a new car with all the modern features.
The only other options to explore are the Greyhound Bus and Amtrak Rail and, as you mentioned, flights. I think that for the trip you are planning to take, a car rental will be your best option. But here's info on the other options for you to check.
Greyhound does offer discounts for online purchases and advance purchases. They also have a Discover Pass, which is for multiple trips. http://www.greyhound.com/home/Default.aspx and http://www.discoverypass.com/discoverypass.usa/step1.aspx
Amtrak also offers a pass. http://www.amtrak.com/servlet/ContentServer?pagename=Amtrak/am2Copy/Simple_Copy_Page&c=am2Copy&cid=1081442673945&ssid=296
The prices of the passes depend upon the number of days the pass is purchased for and (for Amtrak) the travel zones. It looks like both of these passes would be more costly than a rental car. And keep in mind that you don't have the flexibility that you have with a car. Travel by rail or bus will take longer to cover the same distance. And there are some places that you simply cannot get to unless you have a car.
Hope this helps. No matter what, you're going to have a fantastic experience!
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