If you are in the Times Square area and need to catch a train you don't have to run to Grand Central. You can catch a train here at the Subway station at East 42nd street in Time Square. Below is a link of the shedule of the trains listed on the display at Station in the picture.
This is the famous Grand Central Station. Here you can take the Metro North Trains to areas in the Bronx & Westchester County. Also you can catch the MTA Subway Trains to stations in Brooklyn, Queens, Manhattan & Bronx.
In the lower level of Grand Central Station, you can catch a bite to eat from the many different varieties of food. There is enough seats for many people to relax before catching their train to their destination. In this area the tracks leading to some Metro North Trains makes it very easy for you to catch a bite and go.
Renting a car is plain silly if you're planning on staying in the Manhattan area. Between traffic and (the lack of) parking, getting around is easiest if you hail a taxi cab or hop on a double decker tour bus you can find in the Times Square area. They leave at different times throughout the day... you don't have to stay on the bus for the entire ride.. feel free to get off at any stop... catch the next tour bus that stops there or get a cab.
We also used our scooters at Battery Park and took a horse drawn carriage through Central Park.
The New York City transportation system is very efficient and is probably the best way to get around the city.
Here is the main place that most subways & the metro north (new haven & harlem lines) meet. Trains depart from here to Manhattan, Bronx, Westchester, Queens, Brooklyn & Conn.
I feel like it'd take months of subway riding to fully understand the system. Craziness. But convenient.