In 2-3 Days
For a long weekend in New Hampshire, head straight for the White Mountains with a pair of sturdy hiking boots and a tent. New Hampshire is a small state, but try to do too many things in only a couple of days you will waste all of your time travelling between them.
In One Week
Start off by spending a day or so on the seacoast before heading north for a brief stop in Manchester to see the former manufacturing centre and now largest city and cultural and social heart of the state. Continue north on Interstate-93; in the summer veer east towards the Lakes Region for water sports, or otherwise go straight through to the White Mountains. If you have time, consider going west to the Dartmouth College town of Hanover on the Vermont Border, or taking the long drive to the northern tip of the state to search for moose.