Never underestimate ground transport. Amtrak is my preferred means of travel, as it gets me round trip to most of the schools I want to visit for under $150 each. Megabus is even more affordable, but a little less reliable in winter weather. Amtrak always delivers. Greyhound is only for the desperate, and is generally more expensive than Megabus with more hateful customer disservice personnel.
Interview only until you get an acceptance that you really want. Hopefully your list doesn't include any schools you don't want to go to, and has several that you would be very happy with. Once you get one of the preferred ones, turn down any other interviews for schools that you personally rank lower. I know that some people say that you should hold out hope that one of the other schools might turn out to be an even better environment for you, or whatever. And sure, if money isn't an object, see and do everything you can during this process. But if you do have to budget, there really is no reason to see a lot of your B list schools if an A lister lets you in.
EDIT: You asked about where to stay. Youth hostels are pretty sweet and there is usually one in any moderate sized city. AirBnB has nice rooms near most schools. Couchsurfing.com is a way to find free rooms and make some great new friends in the process. Usually when I have couchsurfed, I have offered to buy dinner for my host, but often they have refused and invited me to share their homecooking instead.
If you want a proper hotel, priceline name-your-own-price feature has treated me well. I've stayed in a 4 star in Center City Philly for $50 a night.
It is important to get rest before the interview, and to have a place to put on your nice clothes in the morning, but I have traveled overnight, sleeping on the train, and prettied up in the Amtrak station before going to a professional conference. (You can usually store small luggage in a locker at the station, so you don't have to lug it around. Call ahead to confirm availability!) But that is the extreme cheap method, and I don't recommend it prior to important interviews if you can afford alternatives.