By train
Center City is served by SEPTA trains, both the subway and regional rail. All regional rail lines stop in Center City West, at Suburban Station 16th St and JFK Blvd. If you are coming in from the airport, or Trenton, trains will stop here. Note that Suburban Station only serves SEPTA trains, so if you are coming into Philadelphia on Amtrak, you will disembark at 30th Street Station, just across the Schuylkill River.
SEPTA has only two subway lines, both of which run through Center City. The Market-Frankford Line or the El, because it used to be elevated is a east-west line running underneath Market Street, connecting West Philly to Northeast Philadelphia. Unfortunately, because the subway was constructed at a time when nothing much existed between the Schuylkill River and City Hall, it does not stop in the middle of Center City West. The Broad Street Line is the second subway line, running north-south. However, as Center City is best explored on foot, just get off at City Hall and walk; no need to make a transfer on the subway line unless Center City is not actually your final destination.
A third train line, operated by NJ Transit in New Jersey is called PATCO and comes in from Camden across the Delaware River. The train terminates in Center City West at 15th and Locust.
By Trolley
SEPTA runs trolleys from West Philly and University City into Center City West, which stops at 22nd, 19th and 15th Streets before terminating at City Hall. If you are coming in from the west, trolleys are slower but can be more convenient depending on your final destination. In Center City, trolleys run underground.