lankeela you need a lot of cash to be able to do this effectively.
I did this for a while , following a boyfriend who was on tour over there(!) and I found that the depots are in the seediest parts of town 9 times out of 10. Cheapest accommodation would be hostels (also seedy) so getting to see the states would not necessarily mean seeing an awful lot unless you are loaded.
I was on one bus from Montreal to Fort Lauderdale; I nearly lost the plot by day 4! If you want to get anywhere, the bus only makes 15 minute stops every couple of hours so it can keep moving.
You would have to have an over nite stop or a long wait for the next bus.
Some of the oddest and largest people I ever saw were on those buses, I nearly got stabbed in Memphis and spent 12 hours next to a guy with a knife in his hand......