Verdict first: Athens for a compact, walkable intro to ancient history (3–4 days); Istanbul for vast scale, two continents and world-class food (minimum 4, ideally 5–7 days). Budget: Athens runs €60–100/day, Istanbul €50–90 (stay in Beyoğlu or Karaköy rather than Sultanahmet). Athens: enter the Acropolis at 8am opening, then Monastiraki and Koukaki. Istanbul: Hagia Sophia, Blue Mosque, Topkapi Palace, and the Asian-side Kadıköy (a 20-minute ferry). Direct flights take under 2 hours (€40–80), so a 10-day trip splits cleanly into 5+5.