Amtrak is a costly way of getting to Mystic, but if you don't have a car (or can't rent one), Amtrak is your only other option.
One way prices for adults off peak coach cost $45 and depart Penn Station, NYC (NYP) to Mystic Station, CT (MYS) about 4 times daily. The train ride to Mystic is about 2 hours 40 mins.
Once you've arrived at Mystic you must take a cab to your final destination (either the Aquarium, Seaport or Old Mystic).
Please note that reservations are highly recommended and there is no ticket office at the Mystic Station. Call them at the 800 number to make your reservations or book on-line.
Although this is an option, I'd recommend renting a car which will be less expensive than Amtrak.
One of the things I really like about Mystic is that it is conveniently located at exit 90 off of I-95. Ferni and I will usually get on I-95 from our home which is located near the border of Connecticut and drive down to Mystic in less than 2 hours.
Driving to Mystic is the best way to visit this compact town as you'll want to explore all the areas and there really isn't much in way of transportation here.
By Air : The closest airport to Mystic is the Groton-New London Airport. It is about eight miles away. You can also get to Mystic from larger airports like Bradley Airport in Hartford, CT, which is about 1 1/2 hours away, or the T.F.Green near Providence, RI about 45 minutes away.
By Train : Some Amtrak trains stop each day in Mystic from either Boston or New York Get seats on the shoreline side of the train and get great views.
By Car : The Museum is located on Route 27 approximately one mile south of Interstate 95, Exit 90.
The Mystic Clipper is a gaff-rigged schooner with two masts, three jib sheets, top's'l, and a sky's'l. She's 114" a good size where you can still know that you are subject to the winds, and running before the wind she will make 16 knots. Quite a feeling. The best part is no engine noise, nor foul diesel smoke. Just you, the wind, and the ocean!
Mystic Seaport is about 10 miles east of New London in southeastern Connecticut
From New York City North on Route 95
From right lane, turn right at the light onto Route 27 South
Proceed approximately one mile.
Pass North Parking lot on left (opposite Seamen's Inne)
Mystic Seaport South Parking is second lot on your left and opposite the main entrance Visitor Center. This lot provides the easiest access point to the Museum.
Amtrak trains serve Mystic daily from New York and Boston on a limited schedule. Bus, ferry and air service is available to and from nearby points
Once you arrive you have to walk around the Seaport. I recommend starting with the village and then visiting the ships.
Here is a photo from a parlor of one of the houses in the village.