Based on a 5 day trip to Istanbul in 2007.
Istanbul is an absolutely amazing cities, definitely one of the coolest cities in Europe and in the same league as Rome or Paris. Allow several days to really take in the atmosphere.
I was very lucky to have a Turkish friend as a guide to the city, taking me to both tourist sights and local hangouts. Since I was happily following my guide most of the time, I didn't really compile a list of recommendations (I'll leave that for my next visit to the city).
Just a few random comments: As a huge fan of Ottoman architecture, I just loved Topkapi Palace which is situated in a beautiful park with nice views. Allow at least half a day to explore this place and make sure you don't miss the harem (which closes earlier than the rest of the palace). Visit the Blue Mosque and admire the amazing interior (huge domes painted with elaborate ornaments). Aya Sofya is also very interesting as it combines Ottoman and Byzantine elements. Allow a few hours to explore the Bazaar district, make sure you wander through the small lanes outside of the Grand Bazaar and don't forget to have a look at the Spice Bazaar. Check out Taksim to get an idea of what modern Turkey is like. Explore the Bosphorus and chill along the waterfront in Ortaköy. Last but not least, venture to the Asian side of the city where many Istanbulus live (e.g. take the boat to Kadiköy, check out Bagdad Street).
I can't wait to go back to Istanbul, so this page will grow soon...