Have done several trips of this sort around Eastern USA all prearranged via the net. Start with where you are going to land and if you are self drive then it's much cheaper to take the car back to the same airport. You can then plan a tour route booking hotels where you need them.
Look for hotel that let you cancel as late as 4.00pm on day of arrival, that way you change your mind while there. Also if you arrive at 3.30pm and don't like the hotel you can still cancel and find another.
I usually stay on the outskirts of the bigger cities and use the train to visit them, or start at say in New York and pick up the car later when you need it.

Thanks for that. I've never driven in the States before and the thought of driving in New York makes me shudder! I was wondering if it's possible to buy a rail pass and let the train take the strain? I like your idea of staying on the city outskirts, though, as that will doubtless save a few pennies.

You can purchase various rail passes for the subway on main line trains either on a daily basis or for a number of days. They are cheaper per ticket the more you buy or most of the stations have automatic ticket machines so you can pay as you go. They accept cash or cards. There are daily peak times to avoid otherwise it is quite cheap.

I have in mind a round trip taking in at least New York, Niagara and Washington. I'm thinking of flying into New York and making rail journeys to the key sites, probably utilising taxis to get me to and from the hotels. Does this sound feasible? Will a £1,500 per person budget cover this?

How many of you and how many nights.
The Americans love cars and planes, so passenger trains to and from the major cities are a bit hit and miss.
I personally would hire a car and book one with sat.nav. if you're not very good with a map.
The roads are excellent with lots of motorways for where you are planning to go.

I would certainly avoid driving in New York and Washington if this is your first trip.
Have a look at Amtrak:- let them do the driving
http://www.amtrak.com/servlet/ContentServer?pagename=Amtrak/am2Route/Vertical_Route_Page&c=am2Route&cid=1081355909410&ssid=134

Also Niagara is a long way from New York City- it is at least a 7 hour drive, I don't know if that appeals to you, but you can fly there (into Buffalo) in a couple of hours and you can get around Niagara Falls area on public/tour buses,walking or taxi- be sure to stay on the Canadian side, the US side is somewhat shabby. Buffalo is about 20 miles from the falls hotels so you could taxi there...
I am not sure of rail service to Niagara from NYC but Amtrak can tell you. Or, if you fly/train to Niagara, rent a car there and give the driving a try in a fairly quiet area.