With one week, you'll barely scratch the surface of the city. I think that unless you are absolute history buffs, it's best to only spend 2 or 3 days in the old city. I'd suggest a day or two in Beyoglu, the most cosmopolitan and schizophrenic district in the city. It also has the best restaurants, cafes, and bars in most likely, the entire country. A cruise up the bosophorus is wonderful in sunny weather- at the mouth of the black sea there is a great castle to clamber around on and quite a few pretty villages to check out.
The prince's islands are gorgeous and great fun for a bike ride to work off all the baklava (a good ratio is 3-4 pieces of Baklava per day, btw). The last one, Buyuk Ada tends to be the most crowded, but the old homes are absolutely stunning.
I'd also recommend spending a day in Kadikoy and Moda, which is a nice walk and has an absolutely beautiful park near the shore.
Finally, to maximize your fun: Walk everywhere, eat everything, and make sure you look UP - the best things in the city are hidden, so don't be afraid to poke around.