Tucson, Santa Fe, and Austin would all be great places to visit, but if you aren't driving I fear you'll have a hard time getting around. None of those places has much in the way of public transport aside from buses (and no idea how frequent/comprehensive they are) (but renting a bike might be a good alternative).

You could check out Denver, which does have decent public transport. There's not a ton to do in Denver - I've heard it described as "Indianapolis with mountains," which seems pretty on point. Most of the fun stuff in Denver lies in driving an hour or two into the mountains around it.

Disneyworld/Kennedy Space Center/Orlando? Miami? Philadelphia?

I'm a big fan of Portland, Maine as well for a couple days - but you really need to get a car and drive to fun places.

From New York you could take the train up into the Hudson Valley. There are some charming towns and hotels with great restaurants and gorgeous river/mountain views.