Hello, I live in Bethany, CT about 15 minutes north of New Haven. I do not know a ton about homeschooling communities here, I know a few people who homeschool, for various reasons, one in Wallingford, CT, one here in Bethany, and another in Ansonia. From what they have told me, there is a ton to do here with other homeschoolers. There are groups where the parents sort of switch off in teaching lessons (lots of the parents are certified teachers themselves), and various day trips to science centers or planetariums for lectures and different types of assignments. One of my coworkers who homeschools her middle-school aged son said she had no idea how much there was for homeschooled children/families until she started the process a year ago. She even said that she could easily fill his week without having time to teach at home (of course she doesn't, though).
Anyway, New Haven is great in a lot of ways, Westville is very nice neighborhood. You might also look into Hamden, it is next to New Haven but has more green, larger lots, and while it is still a city, it is a little more suburban. Going north from New Haven is Woodbridge, which is a beautiful town, lots of open space, also a fairly wealthy, well educated community (there are a lot of Yale professionals there). One town north of Woodbridge is my town, Bethany. I am partial to this town, if you are interested in a small town, that once had more horses than people, this would be the place for you. It is very "country" with lots and lots of open space, farm stands and barns but so close to everything in New Haven.
As far as the weather, it is pretty common that each season lives up to it's expectations. Now we are beginning to experience the beautiful fall foliage, but winter is around the corner and was intense last year, lots of snow. There is less snow on the coast, Milford, Branford, West Haven, Stratford, etc. Spring is mild and nice, summers have been okay, lots of humidity but not too too hot. Hope that helps. If you want to ask anymore questions about the area, let me know, you can pm me and Ill give you my email address. Good luck in your research!