My wife and I spent about 6 days in Istanbul, staying in several different hotels, mostly in the Old City. I would say that 2/3 of what is worth seeing in Istanbul would be in the Old City. The Hagia Sofia, various mosques, cisterns, Grand Bazaar are all in a fairly compact area. Walking is a great way to get around, once you get used to very crowded sidewalks. It is a city area of small shopkeepers, no big box stores. Some streets have shops that sell nothing but wedding invitations, or streets that specialize in shoe making. Getting lost can lead to some fascinating places, and it always seemed safe.
The tram that goes through the old city is a great means to get around, easy for tourists to navigate. It is also the best way to cross the Golden Horn to get to Beyoglu. This area has a different feel to it, more the jazz music store, coffee shop, high style clothing boutiques. It involves going up and down hilly streets but we are glad we spent a couple of days just exploring that area. Most Turkish speak some English so it is easy to get around.
Due to the tremendous amount of history and culture in Istanbul (Turkey in general) it is really worthwhile to read up on the Byzantines and the Ottomans. One gets a lot more enjoyment from the sites if you appreciate the history behind it. Istanbul is a unique city, a little European but the Muslim influence makes it more livable I think.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.