The Noonmark Dinner is the Trailways bus stop (this link takes to to Greyhound where you can buy Trailways tickets online) and you can buy tickets at the Noonmark. There is one trip each day, but the schedule changes so check at the Noonmark. Typically the southbound trip leaves around noon. The northbound trip leaves NYC around 10am. The bus goes to/from the Port Authority Bus Terminal in New York on 42nd St and 8th Avenue. It takes 7 to 7.5 hours which is longer than the six hours claimed by Amtrak. The trip is quickest if you can get someone to drive you an hour south to catch the early bus from Warrensburg. It leaves Warrensburg at 7:15am and it takes only 5 hours.
The Amtrak Adirondack is another option; the schedule says it is a 6 hour trip going south and a bit faster going north. There is one train each day, at the Westport Station. North bound leaves Penn Station around 8:20am, arrives at 2pm. South bound leaves Westport around 1:40pm and arrives at 7:40pm. It is a six hour trip, said to be pretty, but it often experiences delays. You can also get to/from Montreal this way but is it painfully slow. It takes about 35 minutes to drive to the station in Westport. You can buy tickets and check arrivals/departures on the Amtrak website. There is no way to rent a car in Westport. You can get a limo service to meet you.
A popular Amtrak option is to drive to the Albany/Rensselear Station at 525 East St in Rensselear and catch a train there. It only takes 2.5 hours. Trains leave every 1-2 hours depending upon the time of day. You can reserve a seat, buy a ticket, and get driving directions to the station on the Amtrak site. Penn Station is at 8th Ave and 31st St. You can rent a car at the Albany station.
You can hire a limo service to take you to New York, Boston, Montreal, etc.
Charter a Plane from Lake Placid
To get to/from New York (or anywhere within about 350 miles) chartering a plane for a family or small group can be a reasonable option.