Time Shouldn't be a Problem
You could make the drive from Buffalo to New York City in a very easy day. Even going by way of the Thousand Islands, Adirondacks, Lakes Champlain and George, and the Hudson River would only take a day and a half to drive. So you've got somewhere north of four full days for sight-seeing and relaxing.
Besides US-20 for a scenic route, Letchworth State Park for a great outdoors site, and Seneca Falls for some history, some other sites to consider while crossing the 'upper' or northern ends of the Finger Lakes would include Montezuma National Wildlife Refuge, state parks on the Finger Lakes such as Seneca Lake and Cayuga Lake, state parks on Lake Ontario such as Selkirk Shores and Sandy Island Beach, historic/educational venues like Ganondagan and places associated with the founding of Mormonism around Palmyra, and formal gardens.
New York's Great Lakes Seaway Trail follows the southern shores of Lake Ontario and the St. Lawrence is another great scenic route as are any of the roads through the Adirondacks and state routes NY-9 and NY-22 down the eastern side of the state. Speaking of the latter, plan on stops at Fort Ticonderoga at least and possibly Crown Point and West Point as well.
Very, very generally I'd plan on a couple of days for each 'shore': two days for the finger lakes, two days for Lake Ontario and the St. Lawrence, and two days for Lake Champlain, Lake George and the Hudson River. That would be a nice relaxed pace with time enough to adjust if you find you want to linger in one spot or skip others.
AZBuck