Am having a really tough time finding walking tours of Trieste...the tourist office website does not address walking tours they may offer..also any other tours: wine tasting, foodie, etc..
Trieste is not a good city for walking tours as such. In the centre, outside of the pedestrian zone, the sidewalks are rather narrow so leading a group of people would be frustrating. Also, it's a very hilly city and the landmarks are not all... grouped so visiting them would require you to be fit and perhaps use a bus. For example, the Miramare castle is some 7 km from the centre (say, 4.5 mi). And even if you decide to do that, you might have to be stepping over people sunbathing on the promenade. :)
And the best would be walking along the amazing Giordano Cottur path (https://www.youtube.com/watch?v=xAyB_nxF74Q), the one above the city (https://www.youtube.com/watch?v=MoX7hXIoruU) and, of course, Strada Napoleonica: https://www.youtube.com/watch?v=Ljsqhk-ALpc But that's all rather far from the centre and would require some transport.
So, my suggestion would be to look for wine tasting options independetly of walking and then ask here for help with the best route to see the landmarks and the kind of things you personally prefer. Usually, you'd see the pedestrian zone (starting with Viale XX Settembre), Ponterosso area, Piaza Unita, walk to the pier for the best view of the waterfront (it goes 100+ metres into the sea so when you turn around...), Teatro Romano (right behind Piazza Unita) and San Giusto (on a small hill above). And the Miramare castle, of course but I also found some other things and places to be worth a look. Depends on your preferences.
Good information..thank you!