P40Power,
It seems that people have covered the topic or your trip pretty well, however I’ll add a few more things. With NY eliminated from your choices, you really don't have too many choices. So your New Haven trip sounds like your best bet.
The New Haven station is not out in the boondocks, but there isn't a whole lot around that area to do. It's more business orientated in that area. However within the station there is a Subway, a Dunkin Donuts, and I think that there is a small deli, should you want a snack.
As for access to the tracks, you shouldn't have any problems as long as you don't do anything silly, like trying to enter an engine without permission. Since Metro North trains arrive and depart hourly on the weekends and even more frequently on weekdays, there are always people coming and going. Then add in the Amtrak arrivals and you got a fair amount of people in the station and on the platforms. So I don't think that you would get hassled.
If you've got one, wear a shirt or cap that is train orientated and that might help too. Or if you really want to play it safe, then just wait for them to post an arrival or departure and then head up to that track for 10 minutes or so. Then go back down and repeat the process.
By the way, one has to go downstairs, then through a long hall and back upstairs to reach the main train station/waiting area. The actual station is also worth checking out at least briefly as it is not a new station. I'm not sure if it's the original New Haven station, but it certainly has some history and is more than 30 years old. So it's worth a quick glance at least.
Now if you can get your dad to let you come to NY, then that opens up a whole new world of options and trips. While I don't know if it will help, your Dad should know that Penn Station is quite safe. There are Amtrak police all around, NY State National Guard troops, and even a few police dogs in the station. So the odds of you encountering any problems, like getting mugged, are probably non-existent, unless you leave the station.
I just arrived on the LSL today into Penn. When I got upstairs, I saw at least three Amtrak police officers, one with a dog, and about five or six National Guard troops (with their machine guns slung over their shoulders) all in the main Amtrak station.
There is even a special Amtrak waiting area that you can wait in until its time to board your train. You have to show an Amtrak ticket to enter this area, so even the bums can't get in there to ask you for money.