I think you might be better (if you can afford it) to try to stay somewhere central, so that you have the option of going back to the hotel in the late afternoon, rather than feeling that you have to be on the go continuously from 9am to 10pm in order to get the most out of it.
I would personally go to NY for 4-5 nights, stay in Manhattan, see things you want to see then fly somewhere beachy, hot and relaxing for a week, then you've got the choice of California (amazing) and Florida, for your chill out bit. (internal flights are CHEAP - DH and I flew across to California from NYC for less than it costs to go on the train from Bham to London). It's hard to find tourist lets in the New York area as they're illegal (I think - or seriously frowned upon), so I think it's better to bite the bullet and stay in a hotel. Stopping in Brooklyn can keep the cost down, but I loved the fact that we were 'living' in the centre of Manhattan for a few days and we had the freedom to walk out for cocktails etc. (We live in the sticks in the UK and if we go out for drinks one of us has to drive or we fork out £30+ for a taxi)
Am hoping to travel to New York next year, we are planning June as have 2dd's 16 and 18. Soooo my dilemma is this, ideally I would like to stay somewhere preferably coastal and commute in so that you can have a day of sight seeing and a day of chilling but I have no idea if this is feasible and what areas to look at (We are thinking of renting a house) or whether it is better to do it in one go and collapse at a beach for a week or so after - a 2 centre holiday.
Any thoughts, websites for advice/planning etc and will weather be bearable?