Dan Shoup is an archaeologist who has worked and lived in Turkey. Recently he passed along his list of top things to do in the city that straddles two continents.
He heads his list with, "(below is my) must see/do list (besides the obvious things, like Ayasofya!)." So you start at The Church of the Holy Wisdom, perhaps better known as Hagia Sophia (Ayasofya in Turkish), and then follow the list below.
A cruise along the Bosphorus can give you the feel of the ancient wealth of Istanbul in the shortest time--you can do it in as short a time as an hour and a half.
Viator offers a 4 hour tour of the 17th century Spice Bazaar combined with a Bosphorus Cruise (book direct).
The Link goes to a very well done virtual tour of the Süleymaniye Mosque. You'll want to see it yourself, of course. Pixels cannot do it justice.
Lots of travelers complain that the Egyptian Bazaar is becoming a tourist trap, but it's still impressive to walk the aisles despite the influx of jewelery and other tourist junk.
Go north across the Golden Horn and you'll come to the art and night life center of Istanbul.
More night life, grab an outdoor table and order something small.
Take a trip to Istanbul's Asian side; the Byzantine Chalcedon, Kadıköy is a comfortable middle-class neighborhood with bookstores, taverns and tea gardens as well as the opportunity to take a ferry cruise.
Kariye Camii or Chora Churchis considered to be one of the most beautiful examples of Byzantine architecture in the world. It's covered with mosaics and frescos.