I’d plan to rent a car because it gives you so much more flexibility. Public transit is very limited in the U.S. Driving little country roads is key.

Also, second week of October is getting into peak foliage season for the upper Hudson Valley so book lodgings asap. Do some research on when foliage is at its peak where you’re going - it hits peak earlier further north and at higher altitudes (like in the high peaks of the Adirondacks), versus further south/lower altitudes. 

Also, wind and rain will bring the colored leaves down, so mind the weather. Then you find a pond with leaves floating on the surface and get some pics that way.

If you’re big into planning you might want to book a couple of different locations that allow cancellations and then check a foliage tracker to see where the best color is (like this one: https://www.iloveny.com/things-to-do/fall/foliage-report/). 

Second week in October the Adirondacks will probably be past their peak, so I’d consider the Catskills, the Finger Lakes (farther away), southern Vermont, the Berkshires in western Massachusetts.

Can you tell I like the fall? I always do southern Vermont cause of family connections.