Greetings from the northern part of the state.
Here are a few suggestions....
-Call or stop by the local PD and ask them where they shoot. Private? Just a sandpit on someones property? (that's what we do) Suggestions?
-Are you near ANY of the White Mt. National Forest (I think you may be too far south?) You can shoot/hunt in the NF as long as you are not shooting across any trail, are at least fifteen feet off the trail, 1/4 mi. from any trailhead, camp, ranger station, etc. This is NOT written in stone...just "common sense" regulations that a ranger told me a few years ago. I don't know what the true laws are for the NF, but I've been doing it for years and all the rangers say to me is "Hey, what caliber is that??" Most of the level one seasonal employees are not up to speed on all the laws, yet they are the ones you will deal with the most. They defer to the local PD's and SO's in the area for law enforcement. New Hampshire is just plain awesome that way. If it looks like you are set up and legitimately target shooting/hunting you're just as likely to have the officer/trooper say "Mind if I join you?" instead of getting scrutinized.
-Ask at the local village store or small "hang out" for the older crowd (a diner, coffee shop) Ask around. There are PLENTY of people in this state that shoot.
-Your own property: As long as you are 300' from the nearest neighbors house, have a safe backstop, and are not in the "compact zone" e.g. downtown or highly populated areas, then you can shoot w/o a problem. Just bear in mind any local ordinances trump the state for certain laws (noise ordinances for example). Check first for your area.
-Suck it up and join a pay club. I don't, but there are a few in the area. I'm not much of a "people person" when I go shoot (I'm there to shoot, not talk all day), so I like to solo it for the most part. Public ranges make my skin crawl.
-You can shoot in the state parks too, but this isn't too likely, as they are designed around the tourist population and are rife with hiking trails, picknick areas, senic outlooks, etc. I wouldn't recommend it. Our state parks are just small enough that it becomes a problem. For example, you CAN technically shoot in Franconia Notch State Park by law, but how realistic is it? Not at all if you care an ounce about safety. WAY too many people around.
If all else fails, just find some old dirt country back road that dead ends in a cul-de-sac, logging area, etc. and go nuts. I doubt you will be bothered for that, but USE YOUR HEAD when it comes to private property, posted areas, etc. I'm certainly not telling you to break the law, but there are a ton of areas to go that no-one seems to mind either way. That's why I love this state. VERY pro 2A for the most part.
Hope this helps!!