You've got plenty of options in NY and VT. Buy maps of certain ranges, learn how to read the map and start with the smallest elevation gain and work yourself up. Catskills, Adirondacks, Harriman, Taconic Mtns, Green Mtns, Whites...etc.
just day hiking? or backpacking? regardless its all about preparation and thinking ahead. If you think there is a remote chance you need a piece of gear, bring it. day hikes obviously may just be that extra water bottle, but if your going multiple days obviously yours going to want more. and the White Mountains are awesome. IMO VT is entirely overrated. get to the whites