Not all areas of New Hampshire have high speed internet. Some places you can get cable or DSL but not both. Other places satellite or dial up are your only options. You will need to call around to check service.
Most of the internet sites will tell you to call anyway. If you ask around on the forums you can get an idea of what areas have what service.
Great phone option for Free Staters in New Hampshire: Liberty oriented VoIP service!
Verizon is a major phone and internet provider in New Hampshire.
1-800-870-9999
Monday - Friday 8:00am - 6:00pm EST
TDS is another phone and ISP. If you are in a TDS area, you can not get Verizon, and vice versa.
1-888-634-8410
Comcast cable is in some areas, but not all of New Hampshire. Rural areas are generally not served.
1-800-266-2278
DirectTV
With many of the nation-wide companies listed here, you'll get bounced around a bit until you get to someone that handles New Hampshire. You will need to have the full address of your new place to get any real information from them.