If you're staying solely in Manhattan, there are hardly any locations for old-school train spotting, but if you can travel just a few miles out there are better opportunities.
1) If you want to observe a busy commuter railway with frequent 3rd rail EMUs and some diesel-hauled local trains, you could try
LIRR's Jamaica Station in Queens. Getting to Jamaica involves either a short (partly underground) trip by LIRR train from Penn Station, or via the E or J-trains on the NY Subway.
2) If you want to see a variety of suburban, inter-city (North-East Corridor) and possibly freight trains out in the open air, I've heard Elizabeth station, across the Hudson in New Jersey is a good, reasonably safe spot. You need to catch a
NJ Transit train from Penn Station to get to Elizabeth.
3) The trip alongside the river to somewhere like Croton Harmon on Metro-North's Hudson Line is quite scenic and an enjoyable train ride. But there may be less traffic to see on this line compared to some others.
I've done the Croton Harmon & Jamaica options myself (if you'd enjoy an hour at Clapham Junction, you'll enjoy Jamaica too), and planned to visit Elizabeth, but had my plans thwarted at the last minute.
I always find it very interesting to "compare and contrast" rail operations in the north-eastern USA to the way of doing things in UK.
