I don't know Vermont well, since it's relatively remote from me, but the Northeast is where lots of big names got their start.
Vermont has lots of great ice, but not in the summer. There are undoubtedly rock routes too... ah, here we go:
http://www.mountainproject.com/v/vermont/105891603If that doesn't look spectacular enough, you can always look a little farther afield. Remember that Northeastern states are tiny. You won't mind driving to New Hampshire or the Adirondack region of New York for a weekend.
A few areas you should know about, with high concentrations of good climbing, comprehensive guide books, and easy road access:
-Cannon Cliffs (NH): big-wall (1000') trad climbing
-Whitehorse and Cathedral Ledges ( NH): multi-pitch trad, including some run-out slabs
-Rumney (NH): sport climbing
-Shawangunks "Gunks"(NY): Endless trad crags, two to three pitches tall, on incredible rock.
For alpine, Huntington Ravine on Mt Washington, NH is well-known. There are lots of backwoods possibilities in Vermont and New York (Maine too) if you're up for adventure, but you may have trouble finding reliable beta. Mountainproject is often your best source.
Finally, beware of "old school" grading...