Istanbul has been a capital for more than 1,500 years, and it still behaves like one. Constantine refounded the old Greek town of Byzantium as his new Rome in 330, and as Constantinople it ran the Byzantine Empire for over a thousand years. On 29 May 1453 the Ottoman sultan Mehmed II took it after a 53-day siege, and for the next 470 years it ran his empire instead. The Republic moved the capital to Ankara in 1923, but nobody told the city. Fifteen million people, two continents and a strait full of ships make it the biggest and busiest place you'll visit in Europe.
It's also easier than it looks. The famous sights sit close together on one peninsula, the ferries are cheap and are the best way to see the city, and the food is reason enough to come: simit from a cart, fish sandwiches by the water, kebabs from every corner of Turkey and long meyhane dinners with rakı. Five days is the minimum to see the big names without rushing and still have time for the Bosphorus and the Asian side.
This plan keeps each day in one part of the city. Day one is Sultanahmet, with Hagia Sophia, the Blue Mosque and the cistern, day two Topkapı Palace, the Grand Bazaar and Süleymaniye, day three the Bosphorus shore up to Bebek and a sunset on the Asian side, day four the Spice Bazaar, Galata and Beyoğlu, and day five the Byzantine mosaics of Chora, the old Greek and Jewish quarters on the Golden Horn and an evening in Kadıköy. If you have longer, the best trips out of the city come after day five. Istanbul is also the obvious way into Turkey: almost every regional cuisine and tradition in the country has made its way here.
Start on a Sunday or a Tuesday if you can. That way no day lands on a closing day: Topkapı closes on Tuesdays, Dolmabahçe, Istanbul Modern and Pera on Mondays, the Grand Bazaar on Sundays, and Chora takes no visitors on Fridays.