I suspect your biggest problem will be your age. Check the big car rental sites (Budget, Alamo, Hertz and the like) to see if they will actually rent a car to someone under 21. Budget don't appear to, and even between the ages of 21-24 there's a hefty surcharge, additional driver must be over 25 and restrictions on the type of vehicle they will rent you. The exception is picking up in New York, apparently, where drivers aged 18-24 get to pay $110 per day extra!
Assuming you can rent a car, you'll have to factor in a significant one-way drop off cost as I can't imagine you'll be returning the car where you picked it up from. Don't forget to add on the Loss Damage Waiver insurance, and it might be worth accepting any additional top up insurance - they're a litigious lot in the States. I usually pass on any other options (personal accident type stuff is covered by travel insurance anyway, and the 'buy a tank of gas and return it empty' deal looks good until you realise as you return the car that you've just donated half a tank to a big corporation).
When choosing the size of car, remember that you'll have to keep it fueled. I always forget, splash out on the biggest and best I can afford, and wipe myself out financially fuelling the damn thing. Increasingly a factor these days now that gas in the States in considerably more expensive than it was a couple of years ago (though still cheap enough to make us weep). Having said that, remember to choose one with enough luggage space. Not that I think you'll be able to get one, but the clean lines of that Mustang convertable are ruined by the suitcases balanced precariously in what is only marginally a back seat because your spare boots have taken up all the room in the 'trunk'.
One month, especially off the interstates, might be pushing it a bit, but do-able if you like driving. We spent 2 months driving San Francisco to Washington D.C. last year. Lot of driving but enought time to take a few time outs here and there, as well as a somewhat meandering route.
When planning, watch out for maximum rental periods. We couldn't find anyone willing to rent for longer than 30 days or so. Ended up arranging to drop the first car at Dallas and pick up another there for our second month. This is also where we learned that booking in advance over the internet is hugely, massively (I mean seriously) cheaper than just turning up to see what is available on the day.
Don't forget also that at your age you can't legally drink in the States. Maybe not such a bad thing if you're driving...
Sorry to be such a pessimist and stomp all over your dream. Just wanted to point out some of the things you'll need to consider. But don't let it go, the USA was made to be driven and the freedom to go where the day takes you is a wonderful feeling (I got a huge kick and an unexpectedly long days drive in the middle of nowhere following Route 66 in Arizona once).
Good luck.
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