For the NJ train: you can get the train schedule from the NJ Transit website.
I wouldn't worry too much about safety on the train -- it's a commuter train, and people commute to work in NYC from pretty far away. And people take it to/from Philly a lot on weekends as day trips. Last summer my wife and I did a day trip on a Friday. Our train left Manhattan around 10 pm, was pretty full for the first few stops, and didn't thin out until the Princeton students got back to campus. Even the last few stops weren't ghost cars.
I would check out whichever NJT station you pick once there -- situational awareness stuff. They should be OK, but late at night you never know 100%. Each station will have security staff and cameras. We parked our car at the Trenton station, which is in a rathole part of a rough town, and had no problems.
The NYC subway is also safe during day and earlier evening, though it might be sketchier after midnight. If you have any questions about it, ask your waiter or bartender when cashing out. You can always take a cab or Uber back to MSG, go downstairs into Penn Station, and take NJT from there.
The biggest tourist trap in NYC is Times Square (and nearby stuff like the wax museum, Guy Fieri's restaurant, etc.). Almost everything else is aimed at residents and people who work in the city.