One strong suggestion: Try a food tour with Culinary Backstreets. They are not cheap, but you go places and taste things you might not otherwise, and learn a great deal about both culinary history and history in general. The Beyoglu area is the focus of one of their tours (we did two last year); it's the area that was considered the "cosmopolitan" part of the city, with Greeks, Jews and many other groups; while that's much less so, traces remain.
Culinary Backstreets is at http://www.culinarybackstreets.com/category/istanbul/ For both tours, the guides were lively, friendly, and best of all, well-informed and passionate. (PS...I'm not on commission...but they did start a new habit for us of looking for food tours wherever we go.)
Links to my blogs of the two tours on TravelGumbo: the Beyoglu tour and the Two Continents, Two Markets tour
If the weather is warm, and you have the time, you can take a cruise of the Bosphorus on one of the Istanbul ferries. Leaves from the pier at the foot of the Galata Bridge, just across from Beyoglu, and goes up to the edge of the Black Sea with headphone narration, stopping at several points, and then leaving you at the last town on the Asian side for a pleasant lunch before returning to the city. Our Bosphorus cruise blog, with a restaurant recommendation for that lunch. There are many companies offering a Bosphorus cruise, but the one from the Sehir Hatlari public ferry company is cheapest and best, about TL25.