Basically, whatever you do GET OUT OF THE OLD CITY and see Istanbul.
It really depends on your level of palace/historical sight fatigue. For instance, if you've been travelling around for a long time and have seen a lot of historical palaces, cathedrals and etc, you really can just get away with the Blue Mosque and the Aya Sofia, which are right across from each other.
If you are, on the other hand, dying to see some history, Topkapi palace is second only to Al-Hambra in Grenada in my opinion. Furthermore, Dolmabahce is a great example of Baroque Ottoman design. In fact, the walk from Dolmabahce to The Aya Sofia (Follow the coast and then the Tramline!) will give you a great perspective on Ottoman imperial mosques. You'll also be able to get a fish sandwich over the bridge, which is my favorite sandwich in the city.
For everyone, the walk from Taksim to Galata tower is a must-see. Make sure you have enough time to get a beer or seven at some of the local bars and get meze on Nevizade Sokak.
Other sights include Ortakoy, which is a beautiful area by the Bosophorus and directly underneath the first bridge to Asia. If you head up from Besiktas past the bridge, you can stop in a beautiful seaside mosque that has a park and just relax with the city's best view.
Also check out this post for some ideas: Could you suggest a basic 4-day itinerary for Istanbul?
