NYC Public Transportation

You can take your bike on the train. In the bus the bike has to be boxed.
You don't have to pay extra.

The bus and train cost $1.50. To change from bus to train or the other way round you don't have to pay again, that is if you have a metrocard. The card "remembers" that you have just used it. That only works with ONE person per card. Cards are available for $15 for 11 rides. (That is One Ride for Free...)
If you only change trains, buying a $1.50 token is fine. You can stay in the subway as long as you want, change as many times in as many directions as pleases you.

If you have to much stuff to fit through the turnstiles, ask the attendant to open the door for you. Make sure he/she sees you swipe the card.

When waiting for the subway go to one end of the platform, because it's more practical to be in the first or the last car of the train. They are not as crowded as the other ones, and you don't get bothered by people walking from car to car. New Yorkers love doing that, don't ask me why.