Have been to Trieste, but used Venice airport, not Treviso.
Your options would seem to be to get local bus to Treviso station, or express ATVO bus to Mestre ('Venice on Mainland') - check the airport website for links to bus company websites, for times and how to buy tickets (in Trieste tickets are not available on the bus). Most if not all websites suggested have an English option, if you don't speak Italian (look for Union Jack and EN, or ITA and tricolor for alternative options)
From Mestre, Regionale Veloce ('fast regional') trains to Venice about every hour, except long morning gap (based on times from Trieste), also IC and Frecce (HST) trains, note that fares differ.
From Treviso, there are through RV trains to Trieste, the long way round via Udine. Or go via Mestre (busy station if you want to train spot while you wait) or Portogruaro (change there). Again fares will differ based on route and train (type(s)).
Look at trenitalia.com website for times & fares (has english option). If you don't want to buy ticket through it, expect to buy ticket from machine - it will speak English and take both cards and cash - Trenitalia are minimising booking office facilities. I would buy singles myself, don't think there is any advantage in getting a return.
Do not forget to validate your ticket before boarding train.