Bus is usually the cheapest in the US, although on some routes the train comes close.
Each has, as times, offered unlimited travel for a month within specific zones (usually three sections of the US), although I have no idea if any such offer is available now.
Seek out youth hostels to stay in. Contact churches in the area, if your own church's leader would vouch for you (assuming you've got one); I know of people who'll put up a traveling young person of the same faith for free, and feed him, too, if they feel safe. Find out if you've got distant relatives who might welcome you for a night.
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