Trieste is a very elegant little town with a lovely city centre. Has upmarket shops, masses of cafes and restaurants, pedestrian areas, many museums and galleries.
Its easy to catch public transport to outlying attractions like Grotta Giganti and Castello di Miramare - even Ljubljana for the day.
Because I was born there and have been there many times, I appreciate the complexities of Trieste. It's Italian, yet not Italian.
There is a large Slovenian population that continues to be subtly separate. Up in the small towns above the city like Opicina and Aurezina, Slovenian is spoken openly. One of the beauties of this town is that you can have traditional Italian pizza and the freshest seafood every, but you can also eat Slovenian and Austrian food. I usually stuff myself with Bismarcks--the best I've ever tasted since my mother died. I recommend the book "Trieste and the Meaning of Nowhere" by Jan Morris for a history of the city unlike any other.