Your budget will allow you to have your choice in most southern NH towns, and also in some Merrimack Valley towns in MA. Before suggesting a community, I'd like to share a couple of things...
Looking to NH, you should realize that most towns won't have natural gas-your heat choices will be oil, propane or electric (avoid electric!) with wood (either pellet-stove or burning cord wood) being an excellent backup/supplement. The vast majority of homes will be serviced by private well and septic system, not a municipal water supply and sewer service.
Subdivisions of McMansions can be found, but they are in the minority. Homeowners Associations are generally found in newer subdivisions, along with 'protective' covenants, fees and rules about what you can and can't do inside and outside of your home.
The Farmer's Market movement is rapidly growing, and this is a good thing! Here's a PDF to introduce you to some in NH: http://agriculture.nh.gov/publicatio...ersmarkets.pdf
(I have this printed and keep it with me at all times!)
Because you mention chickens, you should know that each town has it's own requirements for the keeping of poultry. Londonderry requires 2 acres, minimum because poultry is lumped in with livestock (pigs, cattle, etc). Many other nearby towns are MUCH more lenient.
Portsmouth is likely to be longer than an hour to Boston, and there is no train. However, Dover, Durham and Exeter have train stations (but not a commuter rail like MBTA). If you end up near the seacoast, the nearest MBTA station would probably be Newburyport, MA. If you are in southern NH, the easiest would probably be Haverhill, MA.
Some towns to consider might include Hampstead, Auburn & Chester (Pinkerton Academy for HS is a big plus). Plaistow or Atkinson (close to MA border, good schools). Bedford & Windham have amenities that your family would probably appreciate (parks, active rec. departments, town pool/beach, etc.) Amherst & Hollis are charming, somewhat more rural. You're in the ballpark for home prices in all of these towns.
For the most services, you might even consider one of our 'cities'. With just over 100,000 people in each, the cities of Manchester and Nashua have suburban areas that could work for your needs and offer the services (trash p/u, water & sewer service, etc). Of course not knowing where you're coming from, what you're accustomed to, it's difficult to make an assessment