To get as far north as Saratoga by train, you'd have to take Amtrak https://www.amtrak.com/home (the commuter rail lines don't go that far north) Ticket are cheaper purchased a couple weeks in advance. The train ride up the Hudson Valley is fairly scenic.
Amtrak trains leave out of Penn Station in Manhattan, which you'd have to get to by cab or public transport.
Another option would be to take the commuter rail lines to someplace like Tarrytown or Poughkeepsie and rent a car there. http://web.mta.info/mnr/html/mnrmap.htm There is a bus (M60) that connects to the 125th Street Metro North Station. Commuter trains are less expensive and don't require reservations.
Driving is certainly possible and you can avoid the worst traffic by heading from Queens through the Bronx to head north. (Routes can be detailed if you chose this option.) When you are doing this (day of week and time of day) makes a difference in the amount traffic.
Not sure about public transit, parking and cabs within Saratoga, hopefully a local will advise.
You might comparison price car rentals at various pick up locations to help you decide.
Edited: 10 July 2016, 08:16