Six-year-old Taron is telling us about his “All About Me” school project. “My best friend is Jovian,” he says. “My best friend,” says six-year-old Jovian, “is Taron.” Which is fortunate, for there are no other pupils at the only school on Little Cayman.

This island of 150 residents, with reputedly some of the best diving in the Caribbean, is a short flight from Grand Cayman, but a world away in every other respect.

Were I to come to the Cayman Islands again, I would head straight out to Little Cayman and its Caribbean cottages along the beach, its quiet roads, its nature reserves, its bikes — and its sharks.

These floated lazily next to the jetty at the Little Cayman Beach Resort; just a couple of harmless nurse sharks, along with an eagle ray drifting past and curious barracudas prowling the hulls of the dive boats. If it’s sun, sea, unspoilt beaches and complete peace you are after, do what the sharks do and mosey on out here.

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The sea life at Grand Cayman is less spectacular, competing as it must with jet skis and tourist boats, and grotesque American cruise liners spewing passengers on to the reef.

True, you will see the stingrays cruising for a feeding at Stingray City, the odd grouper or starfish, and the little black-and-yellow-striped sergeant major fish, but you have to get out farther or go down deeper to discover what else Grand Cayman has.

Get out in a helicopter, for instance, to see the reef scattered below and the astonishing deep blue of the sea. Or go down in a submarine, to look at the bottom of the sea off the capital, George Town. This dive, complete with excruciating commentary delivered at clanging pitch when what you really want is silence, is an expensive novelty at £62.

Surface to the grisly sight of four enormous cruise ships moored in the George Town port. Marvel at the sheer size of the American cruise ship passenger waddling through George Town’s cutesy tourist-town schlock. You will be fleeing back to Cayman Kai before you can say rum punch.

Cayman Kai is the other part of Grand Cayman, the secret bit. You can visit the island in three ways. First, very briefly, as you change planes and fly straight out to Little Cayman. Second, in a party way, by checking in at Seven Mile Beach with its string of luxury resort hotels, Stingers Bar, Sharkey’s Niteclub ... you get the picture.

Or, third, do it in a grown-up way. Because you have style. You have children. You paid off your student loan a long time ago. Cayman Kai, in the north of Grand Cayman, is the “quiet part”: grand villas nestling amid palms, with yachts and motor cruises suckling at the piers. Think Weybridge-on-Sea rather than Castaway.

The water is still and the pool is warm; the air con is cool and the villa’s automatic ice-maker even cooler. Pretend to be a millionaire for a week, and watch your neighbour’s seaplane skim the bay. Or curse his sons as they roar around on their wave riders — luxury does come at a price.

And beware, the Cayman Islands are expensive. Being a tax haven, they have no direct taxation, so all of the taxes are loaded on to goods. It isn’t only the stingrays that can sting: when you hand in 50 American dollars to buy a plastic cup of lemonade, and get handed 38 Cayman dollars in change, you feel it. With a currency pegged at four-fifths of the US dollar, and up to 20 per cent tax on nearly all imported goods, the tourist gets whacked twice.

Make that thrice: you also pay a 10 per cent accommodation tax. And as one of the wealthiest places in the Caribbean, its services cost you, too: $25US an hour for a cleaner for the villa, for instance. On the other hand, it means the locals are well-off, Taron and Jovian get their own school run by the eloquent Miss Veronica, and there is barely a whiff of the crime associated with other Caribbean islands.

The biggest danger you will encounter at Cayman Kai is the angelfish lazing under your jetty, or too many rum punches at the Kaibo yacht club up the road (punch and a cheeseburger for a tenner; live music on the beach some nights). And the villas are not unreasonably priced, especially if you go out of season; perhaps £3,000 a week for eight people, cheaper on Little Cayman.

Unlike the places you might book in some parts of the world, these will be genuinely luxurious: real beachfront, as in right on the beach, not glimpsing it from a side window. They might accept a bit of bargaining, too: Cayman tourism has been hit by the US recession and is also under threat from the imminent opening up of Cuba to American tourists.

With the exception of a trip to the green turtle farm at Boatswain’s Bay, where the children can hold the turtles and snorkel with the fish in the lake, and you can sip beer in the breezy restaurant, I would steer well clear of the south and west of the island. Go kayaking in the mangroves in the north instead.

On a cooler day, pop in to see the iguana-breeding programme. Or hang out at Rum Point, a beautiful beach at the northern point of Grand Cayman, with a good fish restaurant, plus plenty of shade and hammocks. And that $12 lemonade. And, oh, a corporate day out for 50 sales executives on a group raft-building exercise.

Well, that happens on Grand Cayman. It is that kind of place. Avoid doing touristy trips such as Stingray City on the days when the cruise ships dock and 10,000 Americans plop into the water. You can look up the schedules on caymanport.com, so you know when the place is going to be overwhelmed. Or take it from the sharks. They know.

Need to know

Getting there: Alice and Ellie stayed in the five-bedroom Villa Emmanuel in Cayman Kai on Grand Cayman and at Little Cayman Beach Resort (littlecayman.com) with Cayman Villas (caymanvillas.com). Properties to rent from £125 per night.

Daily internal flights to Little Cayman from £94 return.

British Airways (ba.com) flies from Heathrow to Grand Cayman from £583 return. The Turquoise Holiday Company (01494 678400, www.turquoiseholidays.co.uk) offers package deals.

Tourist information: 020-7491 7771, caymanislands.co.uk

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