Three weeks. So if we throw away a couple of days at the beginning and end for logistics, that leaves 17 days of riding. If we assume 60 miles a day, that means a trip of about 1000 miles. So let's say you fly into New York City. I doubt you can get much more of a "big, touristic airport in the east." So let's say you hop on the ACA Atlantic Coast route, using the NYC spur, and ride that up to Brunswick, Maine. Then hop on the Northern Tier across New England to Ticonderoga. Then follow the Adirondack Park Loop down to Albany, NY. Then take the Amtrak train back to NYC. That's probably in the ballpark, a little high perhaps, so you might have to ride 65 miles a day, or there are some shortcuts you could take if you fall behind schedule. You could also use the Philadelphia airport if you prefer.
Another option would be to fly into NYC, take the train up to Albany, do both the Adirondack Park Loop and the Green Mountains Loop, and then take the train from Albany back to NYC.
If you prefer canal riding, there are several nice options, but I'm not qualified to detail them. There are no hills on canal routes but plenty in the Adirondacks and Green Mountains (without being excessively cruel about it).