Are there buses or trolleys from Penn Station to Battery Park?

Answer 1

There are no trolleys in New York City. There are buses, but the bus system is extremely complicated, and the maps and schedules are incomprehensible and virtually useless. It's much less hassle to take the subway. I would not take the bus, unless you cannot take the subway because you are walking disabled (the majority of the subway stations are not wheelchair accessible).

Take the Downtown 1 (of the 1-2-3, the red line) from 34th Street-Penn Station to the end of the line, South Ferry. Battery Park is right across the street from the South Ferry Station.

The 1 is a local train. To make it a shorter trip, you may want to take the Downtown 2 or 3 express trains from 34th Street-Penn Station to Chambers Street. It's only 2 stops: Penn Station ---> 14th Street ---> Chambers Street. Then transfer at Chambers Street to the Downtown 1, 2 stops, to South Ferry (it would be 3 stops, but the Cortlandt Street Station is temporarily closed).