You want the Hudson Line when you're looking online.
Be forewarned - many of the train stations are parking strapped. In Fairfield, the day parking is full by 6am. My DH walks to the train every morning, we've been on the waitlist for a parking permit for almost five years.
I think there is much more parking in Westport for day parking.
Try and ride off peak, otherwise you'll be crammed in with the morning/evning commuters. It's not uncommon to be standing the entire trip if you are on a peak train.
My MIL is taking two of my nieces to NYC this weekend, they are from Hanover, MA. They will drive to my house, leave their car here, and then we'll take them to the train. So much easier than driving and parking in the City, esp. if you've never done it before.
Enjoy NYC!