Grand Central Terminals and Penn Station are the two major terminals that handle Train traffic in the city.
Penn Station located at 32nd street between 7th and 8th avenue (next to Madison Square Garden) has Amtrak trains that travel the domestic US as well as the LIRR the line that connects Manhattan to Long Island. Path Trains also run from New Jersey to Penn Station as well.
LIRR Schedules and maps: www.mta.info/lirr/
PATH train Schedules and maps:www.panynj.gov/path/
Amtrak train schedules: www.amtrak.com/
Grand Central Terminal (Park Ave and 42nd) handles the MetroNorth trains that run upstate NY and Connecticut. There are many shops, restaurants and bars inside the terminals so if you are running late you can usually grab a quick bite to eat or a newspaper or magazine before you jump on the train.
Metro North Schedules and maps: www.mta.info/mnr/
Port Authority Bus Terminal Midtown 8th Ave and 40th Street is the hub for all long distance and commuter bus service in and out of the city of New York. You can get service from companies like Greyhound and Adirondack from this terminal. The Port Authority can be a little rough like any big city bus station so keep that in mind.