Tigger, welcome to Philadelphia.
The best place to see locomotive hauled passenger trains is right in your backyard along Amtrak's Northeast Corridor. It will also provide the greatest variety of power including Amtrak Acela trainsets, AEM-7 and HHP-8 locomotives plus Septa AEM-7, ALP-44 and electric MU equipment of multiple generations. New Jersey Transit diesel powered passenger trains can also be seen between Frankford Junction and 30th Street Station, this service runs to Atlantic City. Conrail Shared Assets also operates freight along this line between Frankford Junction and Holmesburg Junction, which is a great place to see all of the above equipment except for the NJT trains.
Septa offers day passes which allow unlimited travel anywhere on the system, including the rail lines although it would be very difficult to cover all of the lines in one day. If you have several days to do some exploring, a weekly Trailpass may be a better bargain. As far as security, the Amtrak PD keep a pretty tight lock on the NEC and may question you, especially if you are anywhere on Amtrak property, including stations. By all means, do NOT go anywhere near the tracks. This is a high speed section of rail and trespassers are definitely not tolerated. Security is a little less stringent on Septa but you can still be questioned by police or crews.
Once again, welcome and good luck.
Glenn