By MupDay - the planner that does the research for you.
The must-see places in Zakynthos include Volimes Mountain Village, Bochali Viewpoint & Venetian Castle, Keri Caves & Lighthouse, Tsilivi Waterpark, Mizithres Rocks and more. Here is the full list, with a short note on each:

Volimes Mountain Village, Bochali Viewpoint & Venetian Castle, Keri Caves & Lighthouse, Tsilivi Waterpark, Mizithres Rocks, Xigia Sulfur Beach
Any season works - MupDay plans around opening hours and the weather, and rebuilds your days as you add or remove places.
Most travellers spend between two and five days in Zakynthos. MupDay builds a full itinerary for whatever length you pick - two days, three, four or a week - splitting the places across the days by proximity and opening hours, so each day stands on its own.
Pick places in MupDay and it auto-builds a day-by-day route with no zig-zag, weather-aware.
Maps © OpenStreetMap.