Not sure if you mean a month in the sound or a month for the entire trip but here's a few tips to consider. Feel free to ask if there's more you'd like to know.
Leaving Hell's Gate and entering the sound, your best harbors are on the NY side IMO. Biggest and most popular being Port Jefferson with good anchoring but you can also go farther into the harbor and get a mooring or slip close to town if that's what you are into. There are other good harbors prior to that as well, but would need to know a little more what you are looking for. Once past PJ, you might want to consider crossing the sound as there is really only 1 other harbor on the NY side, being Mattituck. Mattituck is great too, but would take some explaining and again not knowing your cruising style I won't waste time. Good harbors with marinas and anchoring in CT at Milford and Branford. Cedar Island Marina in Clinton is a great stop with best lobster rolls anywhere at the Lobster Hut. After that there are great stops in Essex, Noank, Mystic and Saybrook. Once beyond the race, Shelter Island has some great stops as well as Montauk.
Sorry, that probably doesn't help much, but again depends on what you're looking for. Everything is available from spartan to luxury, pools and restaurants vs. quiet anchorages, etc. You can't really go wrong, plenty of exploring to do.