One of your first stops should be Natural Marine Reserve of Miramare Visitor Center. Here, you'll be able to find your bearings and get lots of useful information before you start exploring.
You'll find this tourist information center in Roiano-Gretta-Barcola-Cologna-Scorcola. Once you've done all you can at Natural Marine Reserve of Miramare Visitor Center, head to nearby Piazza Unita d'Italia, Miramare Castle, and Canal Grande di Trieste.