No train from Trieste to Split, unless you want to go hours out of your way inland via Ljubljana and Zagreb. Depending on day of week and time of year, there is a catamaran from Pula to Zadar, and you could get a bus from Zadar to Split. As Barbara suggests, renting a car from a town like Rijeka makes some sense, since you could rent the car in the same country you are dropping it in and avoid a huge drop-off fee for the car. I think I might rent the car in Rovinj instead since that's such a lovely little town and only about an hour's drive from there to Rijeka anyway. Rijeka isn't much of a tourist destination.
If you are not going to rent the car, I might take the bus from Trieste to Rovinj, spend a night there, then bus to Pula, catamaran to Zadar (assuming there IS a catamaran that day!), and spend the next 1-2 nights there depending on the timing. You could spend the night in Pula instead of Rovinj - that would probably be easier logistically - but Pula is not an especially charming city.