For the shuttle bus, see the description under the JFK/LGA/PHL heading below. But train travel typically works fine, as follows.

Follow the signs to the Airtrain, the airport shuttle train from the terminals to the car parks and the main train station. Before boarding the airtrain, get a ticket to Princeton Station from one of the orange-blue NJ Transit ticket machines - train schedules are available here. Typically, the price is $18. You should be able to pay by cash, credit or debit card. Exit the airtrain at the last stop, marked in red on the Airtrain map here.

Take the next NJ Transit train towards Trenton (Northeast Corridor Line, southbound direction). There's usually a train at least every 30 mins. Do not board an Amtrak train - your NJTransit ticket won't be valid! (While Amtrak trains are significantly faster, very few trains stop both at Newark Airport and Princeton Junction. So it's rarely worth taking one.)
Hint: try to stay roughly in the middle of the train, avoiding the cars at either end. This increases your chances of actually making the connection at Princeton Junction.
Get off the train at Princeton Junction - depending on whether your train stops at all intermediate stations or only at some, this will be after approx. 40 to 50 mins.

At Princeton Junction take the two-car (again NJTransit) shuttle train colloquially known as the Dinky to Princeton Station. The train departs from a separate platform located to the right as you leave the Northeast Corridor train, next to the car park. After about 3 mins you arrive here.

Walking to Scully Hall should take about 5 mins. The easiest route is to walk past New South and then along Baker Lane, passing Baker Rink and Bloomberg Hall .

Walking to the Nassau Inn (located on Palmer Square East, to the northeast of Alexander Hall, accross Naussau Street) takes about 10 mins. Walk up University Place, all the way to the junction with Nassau Street at the top left of the campus map, next to Rockefeller College. Then turn right and walk along Nassau Street to Palmer Square.

Alternative: a taxi from Princeton Junction to the center of Princeton usually costs about $10. There's also a taxi stand at the dinky station in Princeton (i.e. Princeton Station), and one on Nassau Street in front of Nassau Hall (near Palmer Square). One of the several taxi companies is PrincetonEconomyTaxi, phone 609-799-9180.

These airports are significantly further away. The easiest option is to take the Olympic Airporter. See here for a schedule; the Nassau Inn stop is right in front of the hotel; the Baker Rink stop is close to Scully Hall. You should reserve in advance, here. (While the schedule only shows the timetable for Newark and JFK, bookings may also be made for the other airports.)

Here's how to travel by public transportation - cumbersome but less pricey:

• JFK (here) and LaGuardia (LGA, here): Follow the instructions on the airport web pages to get to New York, Penn Station, by bus (NYC Airporter, here) or subway. Take NJ transit's Northeast Corridor Line, southbound towards Trenton. Again, you'll leave the train at Princeton Junction and then take the Dinky to Princeton Station.
  Alternative: take the NYC Airporter to Port Authority Bus Station (LGA: 55 mins, $12; JFK: 75 mins, $15). Then take the Coach USA (line 100, link here) directly to Princeton, Palmer Square (the final stop, after about 90 mins, $13.95) The ticket counter is located inside the Port Authority Bus Terminal.
• Philadelphia International Airport (PHL, here): Take SEPTA (schedules here) trains to Trenton (change at Philadelphia 30 Street Station; you can get the ticket on the train from the airport to 30 Street). At Trenton, get the NJ transit's Northeast Corridor Line, northbound train towards NY Penn Station. Leave the train at Princeton Junction and then take the Dinky, as described above.