As far as getting a bank account, the simple answer is "it depends". Banks can legally open an account without an SSN (I've done it!), but many will not do so except under exceptional circumstances. However presuming you're spending the time you're in the US travelling around rather than being at a single address it's unlikely that most banks will allow you to open an account without some form of fixed address.
However there is a good alternative - a "reloadable" Pre-Paid Credit Card. These go under a number of different names such as Serve from American Express and GreenDot and many others who do cards from both Visa and Mastercard. These cards normally cost a few dollars to buy, and otherwise generally have no or very low fees (eg, Serve costs $1/month).
Note that these cards are NOT the same as the pre-paid "gift cards" many places sell. Those generally only come with a fixed amount of credit and can not be recharged, but more importantly if your card is lost or stolen, they can not be replaced like the cards mentioned above.
You don't mention where you are from, but a potentially better option is to look for cards in your home country that do not charge international fees. These used to be very uncommon, but in recent years more and more banks have started offering them.