You would be applying for a visitor visa as an "inactif" - which means you would have no right to work (and you would have to show that you have adequate income to support yourself and your kids). You'll also need health insurance paid up for a year.
The kids would need dependent visas - or possibly student visas, depending on what exactly they are planning on doing. The age of majority in France is 18, though it's possible to remain "attached" to a parent's household if you're a student or have other dependencies. I'm not sure how that will affect their chances of getting a visa, however.
Normally, the address you give when applying for a visa is that of your first stop - it can be a hotel or vacation rental if you're planning on looking for more permanent accommodation on arrival.
This is the page on types of visa from the Boston Consulate:
Long stay visa for visitors - Consulat Général de France à Boston Poke around through the section a bit and check the FAQ for more information on your particular situation. A key question on the visa application form asks for your reason for wanting to live in France. In "unusual" situations like yours (i.e. a retiree with children at or over the age of majority), this question can be crucial as far as granting a visa or not.
Cheers,
Bev