The city of Trieste,as well as most other major Italian cities, has excellent public transportation. They have frequent bus service to various parts of town as well as to nearby communities. There is also inexpensive train service that can take you to other cities in Italy and throughout Europe. (A ticket from Trieste to Venice cost 17 euros—about $20.)  Taxis are also cheap.

Trenitalia has made some recent changes to their ticketing system, which I was not happy about. You used to be able to purchase a ticket for a local train and use it anytime within 30 days, but now you have to buy a ticket for a specific day. You can exchange it, if you decide not to go, but that means spending at least 15 minutes in line at the ticket office. This can only be done 24 hours before departure.