One day
Start in the Icherisheher old city — the Maiden Tower, the Palace of the Shirvanshahs, the caravanserais and ramparts — then walk out to Fountains Square and Nizami Street for the oil-boom architecture and a plov lunch. Spend the afternoon on the boulevard along the Caspian, ride the funicular up to Highland Park and the Flame Towers for the panorama, and stay for the towers' light show and a tea-house evening.