All the Montauk line trains are double-deckers -- they're the only trains (AFAIK) with Diesel engines, and the tracks past Babylon are not electrified.

Other than a very select few that go into Penn Station, you'd probably need to catch one at Jamaica. On a weekend, looks like you could, for example, take an 9:45a or 11:45am from Penn, and change in Jamaica to the 10:10a/12:10p to Montauk, and get out at Babylon (from which there should be a good number of options to return, though most won't be double-deckers).
Also from Jamaica, the 2:41p to Speonk or 5:41p to Patchogue.

Here's the Montauk Branch timetable.
http://web.mta.info/lirr/Timetable/B...taukBranch.pdf
The schedules can be a bit tricky to read because of the transfers (i.e., the J and T notations), but happy to help decipher.

If you end up just going to Babylon, then I'd just use the web-based form to figure options back, unless you really wanted a double-decker back as well, but that may be difficult to do because of the schedule.

ETA: Oh, I just noticed the Hicksville part...
I *think* the Ronkonkoma branch past Ronkonkoma (to the North Fork) is not electrified, and should be double-deckers. Looks like there's a 9:55am from Hicksville to Rononkoma, which then connects to a train that continues out toward Greenport at 10:37am. I suppose you could take it a few stops, but then you'd probably need to figure alternate transportation back to Ronkonkoma/Hicksville because trains on that line only run 2x per day.