How does the property tax work in New Hampshire? If one were to move there where are some good places to look into?
Normally, there is the town property tax (decided by voters in full day town meetings), the county property tax (decided by the state reps in the county), school district property tax (decided by the voters in the school district on school budget voting day) and state school property tax (decided by the state government).
I am single so I rent a room. Renting a room is typically $325 - $450 per month in NH plus maybe a share of some utilities for newbies and $250 - $400 per month plus maybe a share of some utilities if you have lived in NH for a few years and have made several connections. That, of course, includes property taxes.
The town property tax varies from really high in towns where people want lots of government services to very low or zero in towns where people want few services or unincorporated extremely rural areas.
The county property taxes tend to be lower in the rural and conservative leaning counties. All of the state reps in a particular county have a say in county property taxes. While all counties have property taxes, some of the unincorporated extremely rural areas in Coos County don't have to pay them.
School district property taxes are decided by how much the voters in a district decide to spend on school budget day. Typically, the more students per district as a percentage of the population, the higher the school district property taxes. So, for example, the towns in the mountains and Lake Region tend to have less kids so they have lower school district property taxes or in extreme cases, none at all. That isn't true for the largest city in NH; though, as the people of Manchester purposely spend very little on schools compared to other cities and many towns in NH.
The state school property tax was created in response to a NH Supreme Court decision which said that the state government has to help fund government education in NH. It tends to be low and is a scheme where taxes are collected locally via property taxes and slightly redistributed and sent to local school districts to spend.
A few very rural parts of NH don't have property taxes. However, very few people choose to live in these places as the government roads are few and far between and if people want government services, they often have to contract for them with near-by town governments. You want power? Make it yourself. You want water? Dig a well.
Many of the towns in the North County and the Lakes Region have low property taxes (rates and average bills). A few communities in other parts of NH, such as the the Seacoast tend to have low rates but in some of the communities the average home price tends to be high. Generally, people pay higher property tax bills in college towns, towns which border college towns, in the Seacoast and suburban areas near MA and in the suburbs of Manchester and Nashua.
If you want to have lower property taxes bills, your best bets are multi-unit buildings, very low prices homes, very low priced homes in town where the average price of a home is high, avoiding college towns, in the North County or in the Lakes Region.
Perhaps the freest town in the US is Grafton but it does have property taxes. People in Seabook pay the lowest average property tax bills in the Seacoast of NH. Of the larger cities in NH (25,000+), people pay the lowest average property taxes in Manchester and Rochester.
Good places to look into? It depends on what you want. I know people that move up from MA. Some of them start out by keeping their job in MA and moving to a town in NH near the NH/MA border such as Salem, Derry or Nashua. For a single mover coming from a far away state, I recommend renting a room in someone's house or apartment for a few months in Manchester, Dover, Keene, Nashua, Salem, Derry, Rochester, Portsmouth, Concord or somewhere near one of those places and looking for a job and place to live.
Shem could likely give you some advice on southeastern NH as he knows several recent movers to various towns in southeastern NH.