Start out by just visiting your local Humane Society, thats what I did. You can ask the staff lots of questions, they always need some kind of help or donations of items, time or money. I usually take my kids every couple of weeks to socialize with the kittens and some puppies and everyone has fun. We also bring some things like toy mousies or jingle balls. Of course you must beware you will probably like some animals so much you will want to bring them home with you!
One way you can find out what shelters are in your area is to go to www.petfinder.org and do a local search for dogs or cats in NH and the animals that come up will usually have the website links right on the names of the shelters they are at. Good Luck!