Driving is the way to get around Montauk and visit the Point. It's a nice drive through scenic areas and woods to get to the point. We dis see people walking on the road but they must have beach houses close by...it's just too long of a journey from town out to the end.
The train station is a long way (3 miles?) from the town/beach. There is a local bus that goes from East Hampton to the town centre, and provides an alternate way of getting there. Check the timetables though...
If you are visiting NYC, Montauk is a great day trip... in case you're not driving, you'll need to get ready to walk... but before that, this train will take you right till the end of Lond Island.
I'm talking about the Long Island Rail Road, it'll leave you 15 blocks north of the 'downtown' area... and about 5 miles west of the lighthouse of the Mountauk Point Park... it does sound like a lot of walking, and believe me when I say it is, but it's worth every step. A very beatiful place! Once you made it to the lighthouse, just have a seat and relax... the view is gorgeous.
If you don't feel like walking, try to hitch-hike... it's not that difficult to get someone to pick you up.