The whale in front of me is slumbering. It lolls in the water – its terrific form bobbing from side to side with the gentle swish of the ocean’s current. It has a calf with it, a replication in miniature that spy hops and splashes in the shallow bay, thrusting its fins skywards and raising its head from the water so I can see the barnacles encrusting its head like briny jewels.

I’m in the tiny town of De Kelders on South Africa’s Western Cape. From the shoreline here it’s easy to spot some of the hundreds of southern right whales that use these waters to rest and suckle their calves each year. As I cast my eye out to sea I can see about 10 of the beasts, each with a calf, wallowing like sleeping giants in the calm, inky waters. Cape fur seals putter past too, as whale calves breach enthusiastically in the distance – hurling themselves clean out of the water in a series of gymnastic leaps – and every so often, the sound of cetaceans exhaling fills the air like heavy-duty hydraulics.

I’ve come to the Western Cape in the hope of seeing the marine Big Five: whales, sharks, seals, dolphins and penguins. I’m well-versed in big-game safaris, having been lucky enough to witness sleuthing coalitions of cheetahs and marauding prides of lions many times. But so far, most of the marine creatures we’re looking for have proved elusive. Given that the nearby town of Gansbaai describes itself as the Serengeti of the Sea, I’m hoping that’s about to change.

My base for exploring is the haven-like Grootbos Private Nature Reserve. Occupying an area of Fynbos biome stretching across 25 square kilometres, Grootbos practically explodes with a riot of flowers (there’s more plant diversity here apparently than the Amazon rainforest).

Our guide Tiaan is just about the most enthusiastic person I’ve ever met, and revels in taking us on plant safaris around the property to explore its 806 species. “What I like best about plants is that they don’t run away,” he calls from the front of the jeep, as we slalom through lush fields of scabiosa, butterfly bush, sour figs and lobelia. We pass huge clown-like protea, which Tiaan leaps out of the van to sniff lustily, and whizz past plants that look like sweetcorn, cacti and elephant’s ears.

Back at our private villa – a glass-fronted, mahogany and marble beauty with a baby grand piano in the lobby, high-ceilinged master suites and bottles of creamy South African Amarula liqueur on the shelves – I soak up the sunset sea views as whale spouts fill the horizon. It’s spectacularly beautiful, and I don’t think I’ve ever been anywhere quite so peaceful, but I’m itching to get out onto the ocean and into the path of the Big Five.

Next morning, I’m up early to head out with Dyer Island Cruises, a company that focuses on whale watching and conservation, with marine biologists on every boat. The morning is crisp, the air flecked with sea spray, and as we motor out of the harbour we’re immediately greeted with an unusual sight. A pod of Indo-Pacific humpback dolphins surrounds the boat, their pinky-grey bodies glistening in the early morning sunshine and their little snub snouts breaking the water like tiny swords. There are thought to be only about 500 individuals on the entire South African coastline, so it is a privileged encounter indeed (and a big old tick for my Big Five checklist).

What we see next, however, breaks my heart. It’s a mother southern right whale with two calves. At first, I’m delighted to see twins. Until the biologists tell us that twins don’t exist in the whale world – there’s no way a mother could sustain and nurture two babies.

This calf, they say, has been separated from its own mother and has been on its own, and vulnerable, for the past few days. It has now attached itself to this mother whale and her calf in the hopes that they will look after it, and has even been trying to suckle. When it approaches the boat, they tell us it’s checking to see if we’re its mother. My heart aches even more.

But we must plough on, and as thousands-strong flocks of cormorants glide across the water in front of us, we make our way to the mist-cloaked islands around Shark Alley. Dramatically buffeted by gnarly waves, the rocky islands are home to huge densities of seals. Thousands of the creatures bark and waddle and hoot and flop into the water to riotous effect. As we motor past Geyser Rock, great gaggles of them cling to its folds and crags – honking, chattering and draping their fleshy bodies over boulders.

The smell is overpowering, stinging my eyes and nostrils, so I’m slightly relieved when we continue on to the next island, to spot a tiny huddle of African penguins, monochromatic against an ocean of white gulls. There’s only one animal left on my list: sharks. But the biologists aren’t confident. Killer whales, it seems, have decimated the current population of great whites, picking off the predators and leaving their bodies to wash up on nearby beaches. The remaining sharks have scarpered, meaning slim pickings for those, like me, who are desperate to see them.

This change has had a knock-on effect on other animals. The lack of sharks means the seal population has exploded to almost uncontrollable proportions, and they’re getting lazy, hunting penguins for the fish inside their bellies instead of heading out into the deeper waters to hunt for themselves. Such is nature. Hopefully the sharks will return soon and balance will be restored. But my time on sea safari has been enlightening and enrapturing. And four out of five ain’t bad.