There are some problems with buying a car when you get to the US. The main problem is that tax, tag and title take some time to get (about a month for the permanent tag). Most states don't let you drive a car with a temporary tag from another state. The other reason your rental is so expensive is because you have four drivers who are ALL 21-22, and you're from a country that has very different traffic laws. Tell them the oldest of you is the only one who will be driving. Aside from taking buses, you're pretty much stuck paying enormous amounts of money for a rental car.
America doesn't have hostels. This country doesn't cater much to a culture of young, poor travelers. I suggest you completely skip Philly and Miami if you want to stay for cheap and not risk being robbed. Those cities can be fairly dangerous. The word you're looking for is "Motel." American car culture hybridized "motor" and "hotel" to make a name for the many, small and cheap roadside accommodations in our great nation. It's always cheaper to stay near the airport. Avoid motels with bulletproof glass over the check-in counter. That means they expect to be robbed, and so should you.
Honestly, there's more to see and do out west, so you should minimize the time you spend on the eastern seaboard. I'd go New York, Boston, Washington. Then I'd drive from Virginia to Georgia along the Blue Ridge Parkway (unless anyone is especially prone to motion sickness), stopping in Asheville, NC along the way. It's the most beautiful scenery you'll see east of the Mississippi, and Asheville has a really fun downtown area with lots of trendy bars.
There's really no reason to go to Atlanta. It's about as generic as a city can get. If you're going through Georgia on the way to Florida, get on I-95 and go to Savannah instead. It's a way more interesting place. When you get to Florida, drive through Jacksonville but don't stop there. Only go to Orlando if you're planning to go to one of the theme parks. From Orlando, you should go to Key West instead of Miami. From there, you can take a 20-minute flight to the Bahamas.
Keep in mind that Texas is HUGE. I made the mistake of thinking I could drive from Mississippi to west Texas in one day and ended up having to stop and stay the night at a hotel because Texas never ends. Make sure you have a place in mind to stop because there is a lot of empty space in Texas, and many of the towns don't have a motel... or wireless service. Make sure you drive Route 66 for at least one day, and don't forget to visit the pacific northwest. It's beautiful up there. If you're going to the Grand Canyon, consider also visiting some of the other national parks, like Zion and Yosemite.
