Trieste "Ronchi" airport is about 25km north-west of Trieste. From there you can take the APT coach service to the centre of Trieste - Line 51, tickets are available from the blue machine just beyond the arrivals gate (cost €2.35) (APT timetable).
A good option, especially at weekends, is to book a seat on the ATVO service desk at the airport foyer) to Venezia Mestre train station and continue on with the train to Trieste Centrale (the duration is one and a half to two hours).
By train: The main train station is Trieste Centrale. In Italy, train tickets can be purchased from an automatic vending machine or from a sales assistant at the station or online. Information about the train schedules, tickets and price can be found in Trenitalia. Remember to validate your ticket with the yellow machine just before travelling, and note, that most of the long distance trains coming to Trieste pass through Venezia Mestre train station, which might involve changing trains at Mestre.
Take the 38 bus (timetable) from via Galatti next to Piazza Oberdan. Get off the bus at the last stop, right outside the main SISSA gate. Tickets are available from news kiosks: You can buy a single ticket, a carnet of 10 tickets or a monthly pass. Validate your ticket once on board the bus. Takes about 15 minutes. Evening buses returning to Trieste are at 16.30, 17.00, 17.30, 18.00 and so on until 21.30. The last bus is at 22.30.
By car: SISSA is very close by the Obelisc, just outside Opicina, as shown on the following map. When arriving at SISSA, drive through the main gate and then press the buzzer to be allowed through the barrier, and drive down taking a right turn into to the underground car park (the lower level is usually free).
A shuttle service between ICTP and SISSA is available at the following times shown here.