Istanbul straddles two continents and 2,500 years of empire. The Historic Peninsula (Sultanahmet) concentrates the must-sees: Hagia Sophia (now a mosque — dress modestly and remove shoes), the Blue Mosque, Topkapi Palace with its astonishing treasury, and the Underground Cistern. Allow a full day for Topkapi alone.
The Grand Bazaar (Kapali Carsi) has 4,000+ shops — go for the architecture and atmosphere, not necessarily to buy. The Egyptian Spice Market is more manageable. A Bosphorus ferry cruise (the short public ferry from Eminonu to Kanlica is the cheapest option) is essential. Beyoglu's Istiklal Street and Galata Tower, the Chora Church's Byzantine mosaics, and the Asian side's Kadikoy neighborhood complete a rich picture.