Friuli Venezia Giulia region have an efficient public transport system that comprises urban and suburban lines, trains, bici bus, etc.
It possible to reach the main cities as well as tourist most interested places, beaches and mountains by bus, train, bike, etc.
Information on the regional public transport can be found here: https://tplfvg.it/it/.

Trieste Airport is located in Ronchi dei Legionari, 33 km away from Trieste and about 41 km from Udine. It is well connected with buses and trains.  

The Trieste airport train station connects the airport with Trieste and Udine in about 30 min and with the main Italian cities (Venice, Milan, etc.). 

The Trieste airport bus station connects the airport to the main cities in the region (Trieste, Udine, Gorizia) by APT bus line. 

Tickets can be purchased at city bus/coach station or at the airport.

More info on Trieste Airport website.