Hi McAvennie,
I spent a week on the island last year, at a place called Lourdas. The island is only around 20 miles long, so a car hired for a few days will allow you to see it all.
Drogarati cave and Melissini lakes are close by, near to the town of Sami on the eastern side of the island - where much of Captain Corelli's Mandolin was filmed. The caves are fine, but Melissini - a collapsed cavern with traditions and evidence of Pan-worship - is spectacular, with deep crystal clear blue and torquoise waters - but get there for midday, when the sun is directly overhead for best effect.
The roads are typically Greek, so stay away from the rough and sheer edges, and you'll be fine. Argostoli, the main town is good for eating out and so forth.
For pure sightseeing, Fiskardo in the extreme north of the island opposite Ithaca, is beautiful, as befitting and artists' retreat.
On the southern tip of the island (town's name escapes me, but is as resort-ish as this tiny island gets - which is to say 'not very'), the waters hold turtles, which come ashore to lay their eggs in August. I was there in June, and I had the great pleasure of spotting and swimming with one.
Myrtos Beach, on the western side of the northern spur of the island, is the beach featured most prominently in Captain Corelli's island.
To escape the heat, should you wish, drive up Mount Aenos in the island's centre for cool woodlands and spectacular views over Zakynthos.
Finally, trips are available to the mainland, to Olympia, which I visited a few years ago. Worth the trip, but take a spare shirt, as the heat/humidity is likely to be high and uncomfortable.
All in all, highly recommended.