Sultanahmet without a doubt. Virtually everything is within walking distance and right on the doorstep literally for the Ayasofya, the Blue Mosque, Topkapi Palace and a 15 minute stroll to the Grand Bazaar or in the other direction for Galata Bridge and the Egyptian (Mısır) Bazaar.
Easy enough to get a tram and the old one stop subway for Taksim if you want too and also a tram if you want to visit the Sultan's Palace where Ata Turk died.
Have a great time.