By plane
There are several options to reach Trieste by plane. The city itself has an airport, located around 35 km away from the city center, but there are also other options, such as Venice airport, Treviso airport, or Ljubljana airport.
Trieste airport
There are multiple daily connections with Rome operated by ITA airways and a daily connection with Frankfurt operated by Lufthansa. Moreover, there are several low cost flights connecting various European cities.
It is directly connected to the city center with a train (~30 min, https://www.trenitalia.com/en.html). You can also catch a taxi from there, which will cost around 60/70€.
Venice Marco Polo airport
Most airlines fly into Venice airport, including various low cost options.
From Venice airport you can either take a bus directly to Trieste (~2 hrs, https://www.flixbus.com/) or a shuttle bus to Mestre railway station (~20 min, https://www.atvo.it/index.php?lingua=en&area=menustandard&idpadre=23&idmenu=35) and then a train to Trieste centrale (~2 hrs, https://www.trenitalia.com/en.html).
There are also private shuttle services which you can book in advance, which will take you directly from the airport to Trieste city center (https://www.goopti.com/en/).
Treviso airport
Treviso airport offers a wide array of low cost connections for most European cities.
From Treviso airport you can either take a bus to Mestre railway station (~40 min, https://www.atvo.it/index.php?lingua=en&area=menustandard&idpadre=23&idmenu=36) and then a train to Trieste centrale (~2 hrs, https://www.trenitalia.com/en.html), or a bus to Treviso railway station (~15 min) and then a train to Trieste centrale (~2.5 hrs https://www.trenitalia.com/en.html).
As for Venice, there are also private shuttle services which you can book in advance, which will take you directly from the airport to Trieste city center (https://www.goopti.com/en/).
Ljubljana airport
Ljubljana is well connected to many destinations with both regular and low cost options.
From the airport you can take a bus to the city center of Ljubljana and then a train (~2.45 hrs, https://www.trenitalia.com/en.html) or a bus to Trieste Centrale (~1.30 hrs, https://www.flixbus.com/).
Also Ljubljana is connected to Trieste with private shuttle services which you can book in advance (https://www.goopti.com/en/).
By train
Trieste can be reached by train from the main Italian cities. There are, for example, direct links from Venice (~2 hrs), Milano (~4 hrs), Bologna (~3.5 hrs), Rome (~6 hrs). For more information check the websites https://www.trenitalia.com/en.html and https://www.italotreno.it/en.
Trieste is also decently connected by train to main central European cities. You can reach Trieste from Vienna (~8 hrs, also with night trains), Munich (~9 hrs, also with night trains), Ljubljana (~2.30 hrs). For more information see https://www.oebb.at/en/, https://int.bahn.de/en/, and https://www.trenitalia.com/en.html.