Everyone stays in Baku. That is the mistake.

Heydar Aliyev International is thirty minutes from the centre. The visa-on-arrival lane is fast if you have the fee ready; Indians on an e-visa walk through even faster. Check in, leave your bags, and go straight to the Baku Boulevard. It is a landscaped promenade running several kilometres along the Caspian, and the three Flame Towers change colour behind you all evening. Little Venice, a set of small canals cut into the park, runs gondolas until ten. The ride costs almost nothing and the children will insist on it.

Tonight, skip the hotel restaurant. Walk two streets into the old city and find a qutab stall — a thin folded flatbread off the griddle, filled with herbs or pumpkin, and the best two-minute street food in the country. Order the goyerti (herb) version if you are vegetarian.

Morning inside Icherisheher, the UNESCO-listed walled old city. The Maiden Tower and the Palace of the Shirvanshahs are within ten minutes' walk of each other. Nobody knows for certain what the Maiden Tower was built for — observatory, watchtower, fire altar — and that uncertainty is part of its appeal. A private city tour gets you both sites with a guide who can answer the questions the signs cannot, in about four hours.

After lunch, Heydar Aliyev Centre — Zaha Hadid's building with no straight lines anywhere in it, which is either impressive or unsettling depending on your mood. Then the driver takes the Absheron peninsula road: Ateshgah, the caravanserai fire temple with Devanagari and Gurmukhi inscriptions carved by Indian merchants from Multan and Punjab; and Yanar Dag, a hillside of natural gas burning continuously. Both places together take two hours and cost almost nothing at the gate. Skip the souvenir shops outside Yanar Dag.

An hour south along the Caspian coast to Gobustan, where more than 6,000 rock carvings of hunters, boats and dancers were cut into boulders over tens of thousands of years. The national park museum explains the sequence well; allow ninety minutes before the heat builds. Then a Lada 4x4 takes you twenty minutes off-road — nothing ordinary survives the track — to a field of cold mud volcanoes. They bubble grey clay quietly, about a metre at a time, and smell faintly of petroleum. Children love them. Stop at the cliffside Bibi-Heybat mosque on the drive back into Baku, worth five minutes even if you do not go inside.

This is the one day where joining a group tour makes more sense than a private car: the Lada transfer is included, the guide knows the mud-volcano track, and the saving is real. The full-day Gobustan and Absheron combination tour from the city covers Bibi-Heybat, the petroglyphs, the volcanoes, Ateshgah and Yanar Dag in one loop. If you already did Ateshgah and Yanar Dag on day two, ask the operator to drop those stops and finish earlier.

Three and a half hours north-west into the Greater Caucasus, past the Nohur lake with its drowned trees and still green water, to Gabala. The Tufandag cable car climbs in stages to about 1,900 metres. In June and July the air at the top is around 22°C while Baku is 30; in August it is the only cool air in the country. Lunch on the mountain is a Caucasian spread — dolma, saj lamb, fresh tandir bread — before the drive back. The Gabala and Shamakhi combined tour from the city covers the cable car, Yeddi Gozel waterfall and the old Shamakhi capital on the way, which fills the day without overfilling it.

One thing to skip: the 'adventure park' zip-lines near the cable car base are priced for tourists and take longer than advertised to organise. The cable car itself and the waterfall walk are the whole point of the day.

Breakfast and check-out. If the flight allows an hour, Nizami Street is the main shopping arcade — Azerbaijani saffron and sealed tins of caviar both pass Indian customs with a receipt, and both are cheaper here than in any airport. The airport terminal itself is designed as a set of timber cocoons and worth reaching thirty minutes early just to look at.