The majority of campgrounds in New England close around Columbus Day or shortly thereafter. Usually that's before snow (and around the peak or just after the peak of foliage season), but there are no absolute guarantees on that. Columbus Day weekend is perhaps the busiest tourist weekend of the year with leaf peepers, so getting reservations for then would be a good idea; other times they're probably not necessary except for quite popular destinations.
The only major city in New England to absolutely avoid being in the middle of with the motorhome is Boston. I would avoid it with the car as well and take public transit if you want to see the city; driving in Boston is no fun at all, and navigating is its own special challenge. Propane is restricted from the Boston area tunnels. Other significant cities may be better avoided, but are not impossible to get around. Do be aware of low clearances (which are usually marked at least somewhat in advance); they are more common than in the western states.
Vermont state parks have no hookups, and many have pretty small sites and tight access roads (often originally laid out by the CCC). Many are in quite pretty areas; if you like a little hiking, the half dozen or so around Groton State Forest are interconnected by a nice network of trails through the forest. Mt Ascutney also has some nice hiking and great views (and a road up nearly to the summit so you don't have to hike all the way up if you don't care to or are not able to).
I second JoeH's suggestion to check out upstate New York, as well; the finger lakes and Adirondacks have some really lovely places. Letchworth State Park is one of the very nicest state parks in the nation.
The Shelburne Museum is large and has a very eclectic collection centered around folk art (if it's centered around anything); plan on a full day to see it if you stop.
In New Hampshire, the White Mountains are lovely. The Mt Washington Cog Railway is worth taking, though not exactly pocket change. It almost certainly is cold and windy at the top, occasionally very, very windy. The Kancamagus Highway is a neat scenic road through the White Mountains, not fast going but certainly reasonable to take with a motorhome (though some of the many pull-offs may be a little tight).
There are a few relatively major roads to avoid with a motorhome (state routes and such). "Parkways", in New York and Connecticut mainly, with very few isolated exceptions are legally restricted to passenger cars and not at all suitable for RVs due to quite low clearances. There are some mountain passes that are not good for RVs, to varying degrees. VT 17 through the Appalachian Gap (between routes 100 and 116) would best be avoided although it is not technically impassable. VT 108 through Smugglers Notch (between Cambridge and Stowe) is worse, and completely unsuitable for RVs, though it's a very pretty road in a car. Most years at least one or two semi trucks get physically stuck in the notch, wedged between rocks, despite their being forbidden...with some big, obvious signs stating as much.
