My guide Dino is not a man to mince his words. “Coming here and not seeing the palace cellars is like going to Rome and missing the Colosseum.”

I keep quiet about my previous visit to Split. It was only a morning spent in transit, but I’m not sure Dino would forgive the omission. Actually, it’s a blessing that a lot of people do not visit the cellars. Upstairs, in the old town, the narrow lanes are packed with tourists – Split airport broke Croatian records for numbers this year, reflecting the massive and fast-growing popularity of the country. Down here, however, beneath the crowds, it’s cool and quiet. The truth is that the empty vaults of a Roman emperor’s palace do require a guide to bring them to life, and in particular, a Dino. He does it with extra zest.

“What did the Romans ever do for us? They discovered bulimia! Tell it to the billionaires out there on Split waterfront in their super-yachts. Look in here, this was the triclinium, where the rich would stuff themselves then go over there into the vomitorium and throw it all back up, ready for more. That’s why when they found skulls down here the rich had no teeth – worn away by stomach acid – but the guy tickling the emperor’s throat with a peacock feather, he had great teeth.”

There is no doubt that Split is unique. No other city on Earth can claim to be built inside a Roman palace, and not only that, one that is reasonably well-preserved. Emperor Diocletian had it constructed around the turn of the fourth century, then retired to it in 305 AD, thus avoiding the violent fate of many other Roman rulers. Dino is very clear on what happened after the empire fell. “For centuries the people dumped all their shit down here. Right up until 1952. When they dug it all out they filled three museums with stuff: I mean with artefacts, not shit.”

What also happened was that the inhabitants recycled anything handy for building. Diocletian’s mausoleum became the cathedral of St Domnius, the Temple of Venus was converted into a baptistry. Dino points out the face of an Egyptian sphinx, brought from Egypt by Diocletian to decorate his new Grand Designs waterfront apartment, and now part of an otherwise nondescript house in a narrow lane hung with someone’s washing. The wonder of Split is the way modern living is insinuated into the fabric of an ancient palace: original columns still adorn the waterfront facade, the spaces between filled with shuttered windows; an arch over a Roman thoroughfare buries itself in someone’s bedroom; the guardhouse over the golden gate is cannibalised as a chapel – the narrowest in the world so they claim.

Outside the gate we approach a massive statue and touch the brass toe of 10th-century Croatian hero Gregory of Nin. It is said to bring good luck. I ask Dino how the city’s history has influenced the Split character. “Tell us to go right and we’ll go left. We never eat breakfast, maybe a little marenda – what you call brunch. When we talk – Oh my God! – it sounds like people walking over a bed of chewing gum, all twangy. We love sports and arts because that saves us from hard work. And we are wary of emperors. When [hometown boy] Goran Ivanišević won Wimbledon in 2001, 100,000 people came to greet him. Next day he couldn’t get a table for coffee on the waterfront. That’s Split.”

But it is on the subject of food that Dino gets most animated about food. He is a chef and loves it. “In the mornings, go to the Green market by the palace to buy soparnik.” That’s a snack made of layers of dough and chard with plenty of olive oil. The market is also the spot to stock up on home-grown vegetables and superb fruit: big fat peaches, apricots and plums, piles of tiny dried figs and yards of plaited garlic bulbs, home-made cheeses and cakes.

Dino bumps into an old friend, Ivana Gamulan, in the street. She is the force behind one of Split’s most famous restaurants, Villa Spiza, which looks like a simple hole-in-the-wall canteen with potential diners waiting in the alleyway outside hoping that the food won’t run out before they find a stool to perch on. The staple is brudeto, a fish stew based on whatever Ivana finds each morning in the fish market (worth a peek before 10am). As the day wears on, dishes get crossed off the menu and by 8pm there aren’t many choices left.

Nearby, up some steps inside the palace wall, is Restaurant Dioklecijan, where Dino swears the best prosciutto in all Croatia is served. Once again it’s nothing fancy, none of the swank you find in the seafront cafes, and fellow diners might be buskers and ferrymen. Lunch is the big meal, often taken around three, often washed down with a glass of local wine: a white from Hvar or a red from Pelješac peninsula near Dubrovnik.

A tip from Dino leads me to my own favourite, Zlatna Ribica, next to the fish market, which does excellent pržene lignje (fried squid), and miješana riba, a plate of seafood, that will feed two for under £8.

By now I’m thinking I need a respite from the crowds so my daughter Maddy and I head for the Marjan, a forested peninsula within walking distance of the town. There are good beaches here and shady traffic-free walks up to a high point. For us, however, the attraction is the cliffs, and after directions from a helpful local climber, we do a few climbs. The stunning views over the sea bring gasps from Maddy; the discovery that one route leaves precisely one metre of available rope brings a gasp of relief from me.

Our trip is almost done now, but there is the excitement of driving down to Split’s waterfront and finding our way on to the ferry to the Italian port of Ancona, hopefully a handy shortcut on the long drive home and a last chance to see the clear azure waters of the Adriatic – one of my most abiding memories of a wonderful journey.

Discovering Croatia: why everyone’s loving it

This summer has seen Croatia become even more popular than previous years. The exchange rate helps. The miraculously clear seas help. The people, food and history are huge pluses. It has the vivacity of Italy and the complex depth of the Balkans wrapped up in some tasty dish – like the improbably delicious prawns with strawberries I ate in Mošćenička Draga.

• The trip was provided by Kirker Holidays (020-7593 2283, kirkerholidays.com); three nights B&B at the Marmont Hotel, from £639pp including flights from Gatwick, transfers and concierge services. Ferry from Split to Ancona provided by Direct Ferries (08718 900 900, directferries.co.uk). Further information on Croatia at croatia.hr, and on Split at split.hr

To find discount codes for easyJet, Thomas Cook and other leading travel providers, visit discountcode.theguardian.com