If you are taking Amtrak, you will need to change trains and stations in Boston. Amtrak from other destinations (primarily New York) comes to South Station. The Downeaster to Maine (also Amtrak) leaves from North Station. You'd have to take a taxi or public transit to get from South Station to North Station if you want to continue your journey by train. The Downeaster only goes as far as Brunswick so that's where you would be starting your vacation, unless you got off in Portland (not a bad idea), spend a few days in our lovely small city, and then head further afield in a rental vehicle.

Alternatively, you could get on a bus (Concord Coach) at South Station and come to Maine that way. The busses are very clean and run frequently. You could take the bus to Portland or continue on to Bangor. From Bangor, you'd need to rent a car (or truck) and then you could head to Acadia National Park (Mt. Desert island), Schoodic Point, or a million other coastal locations. Or, you could head inland to the mountains and visit Baxter State Park, Katahdin Woods and Mountains National Monument, go rafting on the Penobscot or again, a few dozen other options. Maine is a big beautiful state with tons to see and do.

I think June is a wonderful month to visit. Things are open but the state is not overrun with tourists since they don't tend to arrive until July and August. September and October are also beautiful. And of course the summer is fine, just be prepared for more people. I'd avoid March and April is also pretty iffy.