I spent part of my junior semester study abroad program in Turkey (in 2007). I was studying ancient Greek and Roman art so my suggestions are mostly based on that. I went to some pretty remote sites; but one of our "jumping off points" was Kusadasi. It is a resort town on the aegean coast where a lot of cruise ships hit. This can be a good and a bad thing. It has a pretty lively night life and there are lots of shops and things to look at within the city, but it can also be very crowded. From Kusadasi we visited Didyma, Priene, Pergamon, and Troy. They are all kind of out of the way, but great places to visit if you want to see ancient Hellenistic temple ruins while in Turkey.
Bursa - south of Istanbul - was the first capital of the Ottoman Empire and also has thermal spas that have been in use since the Roman Empire. There are also several mosques there worth seeing. I remember visiting the "Green" Mosque. I remember shopping being particularly good there with a lot of local craftsmen and a big silk trade.
In Istanbul, we did go to Hagia Sophia. You could spend a lot of time in there looking at the architecture/frescoes/ mosaics/ learning about how it went from a christian church to a mosque to a museum... it is a very interesting place to visit. I would also visit the Blue Mosque in Istanbul - impressive grounds and interior and very interesting to see a functioning mosque with people praying in it(make sure your knees, arms, and head are covered - if not, they loan you stylish blue pieces of cloth to wrap around yourself). I was in Turkey during Ramadan, so unfortunately, all of the bazars and most of the shops were closed - so I can't tell you anything about what the grand bazar is like, but it is one place I wish I had been able to visit. Be ready for people to be in your face and try to push merchandise/ restaurants on you and when making purchases at stalls or in the bazar be ready to haggle on prices.
Hope this is helpful! My recollection of the places we visited is kind of spotty as it was almost 6 years ago (yikes, I feel old!)