There are 3 daily flights on American (PSA) between New Haven and Philadelphia. To me, that’s the most convenient, one layover from say SFO-PHL/PHL-HVN and a short 10 min $10 Uber ride.
Hartford has more flights, but no direct flights from the Bay Area (anymore) so you’re looking at a layover and a one-hour drive, a $60 or so Uber ride.
Flying direct to EWR and navigating a couple of trains for 3+ hours and then a short Uber ride doesn’t seem very appealing. From a time standpoint though you could Uber from EWR to Yale and get there in probably 1 hour 45 min but would probably cost $100-150.
When my daughter was an undergraduate at Yale a few years ago, she would fly to LGA, take a taxi to Grand Central and get on one of the almost hourly Metro North trains to New Haven’s Union Station–from Union Station would be a Yale student shuttle to take her to the Central Campus. (Once she flew to Newark and did the same thing.) The cab ride to Grand Central was the most expensive part but not out of line. It was cheaper than Amtrak and far more flexible. Metro North has so many more choices and is very cheap. Grand Central is not the bus depot experience that Penn/Amtrak is. With all the LGA construction, the taxi trip to and from Grand Central might be more difficult these days but I am not certain.
Another option is to take the Connecticut Limo or Go Airport shuttle from whatever airport to Central Campus. Those go over Interstate 95 which is subject to peak hour tie ups and it often has stops along the way to drop off fellow passengers–these services usually build in 3 hours from pick up to your flight time. But it is “one stop shopping” if you want to avoid transfers and is the easiest choice for someone not yet familiar with where are places like Union Station, Grand Central, how to transit from the airport to the train station, etc.
Problem of flying into Phily is that by time you get into Phily there is only one flight you can get into New Haven, it’s a prop plane that is often delayed or canceled.
well, from SFO there are 3 red-eye flights to Philly every night so that can mitigate the possible missing of connections, since there are 3 flights from Philly to New Haven.
I just find having to navigate train stations with a lot of luggage is very awkward. From SFO, if I were doing this without thinking about budget I probably would fly direct to EWR and simply use Uber or Lyft or some limo service.
My idea of budget-free travel would be a private jet into New Haven , lol!!
for all practical purposes NHaven doesn’t really have an airport. Hartford is the closest, if you’re not counting the teensie one called Tweed.
Phila is too far when you have closer options. And in true NYC fashion, some think navigating a multi step process is no trouble at all. But it dwarfs the inconvenience of changing planes, lol.
EWR to New Haven is a 2+ hour drive (plus city traffic jams.) From Providence airport, 90 minutes. (Or taxi into Prov and every train out of Boston stops in New Haven.) From LGA, 90 mins, and JFK is 2.25 hours, plus city congestion for both if those.
Keep what you can as simple as you can.
I’d call the college and ask them the surest thing. Theyll know. Admission staff flies all the time.
Or listen to the people on this forum who have made this trip often–locals who have done this often. If you have few bags, and flying nowadays most people don’t carry much, the process outlined above is not bad –
NJ transit to Penn station
Subway Red 1 train one stop to 42 and change trains for S to Grand Central
MetroNorth leaves about every hour to New Haven.
If you have lots of luggage or are not willing/able for other reasons to do that, then take the Amtrak directly to New Haven. Easy peasy lemon squeezy.
Dustyfeathers, you’re proving my point that what locals endure without a second thought can be mighty complicated for a newbie. 
Yes, let’s get back to EWR/Amtrak. And Newark is an easy airport to navigate. Later, OP’s D can figure out the other options or tag along with a savvy friend.
Yes, I also pointed out the Amtrak option.
Yes, DF, agreeing with you on that.
New Haven Airport is a great option! Sometimes, it can be surprisingly cheap as compared to Hartford.
Just make sure you specify NEW YORK Penn Station or NEWARK Penn Station when you buy your ticket. Depending on which one you want.
Oh excellent advice!!! So easy to make that slip up.
Is driving not an option? It would usually take around 2 hours and can be more convenient.