Many flights into New Hampshire land at Manchester Airport. Busses run between the airport and downtown Manchester. International flights land at Logan Airport in Boston, about an hour south of Manchester. Busses run between Boston and the Manchester Airport or other areas in the state.
Train service in New Hampshire is limited. The only two Amtrak stops in the state are at Portsmouth on the Downeaster line and Claremont on the Vermonter line.
The best way to get around is by car. Other means can get you into the state but once you are there, public transit is limited and most of the sites are too far apart to walk between. Besides that, driving is the best way to see the scenery.