There are several companies that do boat trips around the island, but the best choice is run by a guy called Peter. I know this sounds dodgy, but it isn't; I just can't remember the name of his company. He spends most of his time touting for business outside, or just nearby, from the Iliessa Beach Hotel in Argassi. His boat has all the appropriate safety equipment, and they take safety very seriously. If it's rough weather then you simply don't go out.
We took a trip and it was wonderful. You leave the port of Zakynthos Town at around 9am. Heading up the island, it's a bit rough for a while, but not too bad, and once you reach the Blue Caves and Cape Schinari it's much calmer. The Blue Caves aren't blue at all, but they appear so because of the reflecting light on the rocks and everything around them from the water. You can take a smaller boat right up to the caves from Aghios Nikolaos, in the north of the island.
Next you go around the top of the island and head to the famous Shipwreck Beach, where a rusty ship is marooned on the sand. Here the boat moors for a while, allowing a swim or a dash to the beach. (The beach is only accessible by sea). Next it's on to Porto Vromi and along the coast past beaches, coves and caves, before arriving at the caves of Keri. The boat gets in close for you to examine the caves and the large one - Papanikolis - is supposed to have hidden a WW2 sub ! It was too rough to swim here and so we headed onto Laganas and the turtle shaped island of Marathonisi. Here is the best chance you have of seeing turtles and the boats have to slow down.
Then you continue on around the cape of Gerakas for more views of the beaches. We had our second swim-stop here, but it's the captain's decision where to stop, which he makes based on weather conditions. We travelled on past Argassi and back into the port at Zakynthos Town to find it was almost 6pm. A great day trip.
My advice would be to bring plenty of sunblock, because the windburn is bad enough...Also bring plenty of water, a hat and sunglasses, and some food. The food on the boat is more expensive than in the local supermarket, but still good value.
Book your trip for partway through your stay - that way if it is cancelled you have more chance of rescheduling instead of skipping it completely.