I'm not big on museums either, so when I go to Trieste, I usually either go walking along the Strada Napoleonica that Andrew mentioned, or walk on the clifftop Rilke Path between Duino and Sistiana. I also enjoy visiting the nearby town of Muggia and it's cute little Venetian town centre. I always end the day watching the sunset from the seafront in the old town, followed by an aperitivo in one of the back streets. My local friends usually take me to one of the many excellent osmiza (spelled also as osmica), inns serving local food run by the Slovene communities in the Karstic countryside, but on a day trip you would probably want to stick closer to the city.
In terms of sights, That New York Times article has some pretty good suggestions on what to visit in Trieste, though. I would probably add the Risiera di San Sabba, a turn-of-the-century industrial building that was turned into a Nazi concentration camp for political prisoners and Jews from Trieste and neighbouring regions, as well as the Chiesa di Santo Spiridione, a Serbian Orthodox church by the Canal Grande, a testament to Trieste's multicultural history.