You don't give your timeframe, of if you plan to camp, hit motels or what, but having grown up in Ithaca and spending most of the first 33 years of life in that area, (before I became a cyclist as an adult) I would strongly suggest avoiding Hwy. 20. The hills east of Cazenovia would challenge even the most ardent hill-climber. (of course the descents would be another matter!
Google Maps will show you a bike route that keeps you out of NYC
and crosses the Hudson on the Bear Mountain Bridge, goes through Yorktown/Mt. Kisco, and into Stamford from the north. To my way of thinking, that would be a lot better than crossing the GW, going through Manhattan and Da Bronx, particularly on your first tour. (Although I admit to a certain amount of intrigue about riding across the GWB. I've only ever driven or ridden a bus that way before.) You will also find more camping opportunities on the Google Maps route, since it takes you through/past a number of state parks and forests, and the Pocono Mtns. in PA. Tweek it to the north a bit, (you can click and drag the suggested route) and it can take you through the Catskills. Sounds like a great, scenic ride either way.
Also...start off with some shorter trips to get ready. There are a number of NYS Parks within 50 miles or so of Ithaca (Not counting, of course, the ones that surround Ithaca.) such as Watkins Glen, Fillmore Glen, Green Lakes, Cayuga Lake (near Seneca Falls; an excellent ride because you can ride up one side of the lake and down the other.) and the Finger Lakes National Forest near Hector. They make great destinations, and you can stay the night, then ride back, and they are not too far from base if you have to bail.
Good luck, have a great ride. Wish I was 33 again...I'd do at least the part to the Poconos/Catskills!