Manhattan offers Subway , Busses and Taxi as means of public transportation. You should make your choice depending on your destination and the time of the day. During the day the Subway is safe and convenient if you are going downtown, for example to see the Ellis Island or the Statue of Liberty.
Buses are convenient to get across town.
Free bus maps are available at the clerk booths at the subway stations. Although Subway and Buses run 24 hours a day its safer and faster to take a taxi later at night. The yellow cabs can be found in abundance all over the city. It's a considerably more expensive way of commuting, but it provides privacy and door to door service. Keep in mind though, that NYC is famous for it's daytime gridlocks and it would take you a very long time to get anywhere if you choose the on-ground transportation.