Some of the villages above Trieste are nice for a wander. Gostilna Brin and Ristorante Carso are two good places to eat I remember. There are also the 'osmica' which do good local food. Quite interesting to see a south slavic language in Italian villages, hard to imagine it was right on the border of a country that stretched all the way down to southern Macedonia.
In Trieste itself Buffet da Pepi is a bit touristy now but famous for large portions of sausage and sauerkraut and frothy beers. Good fun.