Istanbul is one of the most unique cities in the world and with two continents of treasures, three days will hardly do it justice. A week will give you time to enjoy the sights, sounds, and smells with a little leisure. Make sure to see the main sites like Topkapı, Aya Sofya, the Blue Mosque, the palaces, and the bazaars, then seek out more of what you like: there are plenty more Ottoman mosques and Byzantine monuments. Or you can just chill out, çay (tea) in hand, by the waters of the Bosphorus.
Here’s one way to plan your visit if time is limited: Start off as if riding a whirlwind, visiting the Blue Mosque and Aya Sofya to get a sense of the grandeur of the Byzantine and Ottoman empires. In the afternoon, venture out of Sultanahmet to the Kariye Müzesi to see amazing frescoes and mosaics from the Byzantine era, then visit the beautiful Süleymaniye Mosque on your way back. Next, head to the Grand Bazaar, where you can wander until closing time at 7 pm. If you're staying in Sultanahmet, take an evening stroll to see the floodlit domes and minarets of the Blue Mosque and Aya Sofya.
You could spend your second morning at Topkapı Palace, having lunch at one of the restaurants in the compound or just outside the gates so that in the afternoon you can easily visit the Istanbul Archaeology Museums in the palace forecourt and/or the nearby Yerebatan Sarnıcı (Basilica Cistern). You'll have probably reached your quota of treasures-and-antiquities appreciation, so next immerse yourself in Istanbul street life. Head to Tünel Square in Beyoğlu, above the Golden Horn, and follow İstiklal Caddesi through Galatasaray Meydanı, stopping at the Fish Market and Çiçek Pasajı (Flower Passage), to Taksim Square, the center of the modern city.
With another day, you could see more of the Byzantine period, ogling the exquisite mosaics in the Mosaic Museum. For even more evidence of Ottoman power, head to the lower Bosphorus to see Dolmabahçe Palace, the naval and military museums, and Yıldız Parkı.
If you want to see the Bosphorus but don't have a whole day, either take a short, privately run cruise, or take an evening commuter ferry to relax after a hard day's sightseeing.
If you have more time, you can enjoy Istanbul with a little leisure. Think in terms of neighborhoods: you'll want to spend at least three days in the Old City, seeing the Byzantine and Ottoman monuments, and a day heading out to the less-visited western districts. Plan a full day around Karaköy, Beşiktaş, and Beyoğlu, checking out the Istanbul Modern or the Pera Museum and wandering around these interesting neighborhoods, and a day cruising up the Bosphorus. You won't have trouble filling additional time: maybe visit the Rüstem Paşa Camii in Eminönü to see its gorgeous tiles, then return to Sultanahmet to compare them to those in the Blue Mosque. Or take another swing through the Grand Bazaar and perhaps a trip to the Princes' Islands. Or just wander: few cities reward walkers more amply.
The Museum Pass Istanbul allows single entry into eight state-run museums over a period of 72 hours; the five most significant of which are the Aya Sofya, Topkapı Palace (including the Harem), the Istanbul Archaeological Museums, the Mosaic Museum, and the Kariye Müzesi. At 85 TL, it’s a significant savings over the cost of paying for all these museums individually, and it also includes discounts off entrance fees at the private Rahmi M. Koç and Sabancı museums and 10% off purchases at all of these museums’ gift shops. The other advantage of the museum pass is it allows you to bypass entrance lines, which will save time—a commodity you’ll certainly need if you’re going to pack in so many museums in just three days.