Maiden Tower, the Palace of the Shirvanshahs and the lanes of Icherisheher, with tea in a caravanserai.
Day 2
Sea and skyline
The Carpet Museum, a walk on the Boulevard, then the funicular to Highland Park for the Flame Towers at dusk.
Day 3
Zaha Hadid
Morning at the Heydar Aliyev Center inside and out, lunch in the centre, Nizami Street and Fountain Square in the evening.
Day 4
Gobustan
Rock art and the mud volcanoes in the morning, with Bibi-Heybat Mosque on the coast road back.
Day 5
Fire
Ateshgah and Yanar Dag on the Absheron Peninsula; go late afternoon to see the flames after dark.
Day 6+
Snow
A long day or an overnight at Shahdag for skiing, lessons or just snow for the children.
Honest thought: the wind makes 7°C feel much colder. Plan indoor stops between walks, and treat Shahdag as an overnight if you can; six hours in a car is a lot for one day.