There is a map that is available at some NH rest areas that have the major snowmobile trails on it state wide put out by the NH snowmobile assoc. These maps give you a "general" idea of where the trails are and where they go however for navigating they aren't terribly great. Each club produces more detailed maps which are better however these are not free- the proceeds do help support the clubs.
Since you probably aren't aware, the snowmobile trails are mostly crossing private property and by club member volunteers. Each club maintains an area of trail, clearing them in the fall and grooming them in the winter time as well as signing them for the season. Although it may be legal to ride wheeled ATVs on the snowmobile trails in the winter time, you will more than likely find that snowmobilers will not be to happy to see you riding on them as the ATVs tear up the surface of the trails and wreck all the hard work the groomers do to keep them smooth and flat. If you're going to ride an ATV on the trails I'd suggest you put tracks on it.
Trails that are used for snowmobiling are not necessarily open for summer time ATV use. In fact there are a fraction of trails available for legal ATV use versus snowmobiles although it should be noted that there are areas you can ride in the summer and it seems that more terrain is opening up.
If you are primarily looking to use your toy in the winter time I would suggest you get a snowmobile. Far better in the snow especially if you really want to go out on the trails and explore or better yet go on an epic all day ride.
NH online snowmobile map:
http://www.jimapco.com/maproom/snowmobile/nh/
NH ATV trail map (see linked PDF)
https://www.nhstateparks.org/Activit...formation.aspx
Hope that helps.