I've been twice for a week, once with family, once with MrNC. Both times stayed in small hotels on the edges of Sultanahmet. The trams were great though wasted half a day second time trying to buy a pass that didn't exist, so we just got single tickets. You could easily fill a week, though as the city was flooding we left 12 hours early last time (November and 3000 homes flooded. The rain wasn't a problem until then though).
Lots of great food - ask the staff in the mosques or museums for recs. There's one traditional Turkish baths that lets men and women in together, which was great for me and MrNC - up the hill by the red mosque. Previously went to the main touristy ones- mine reeked of fish being cooked on camping stoves in the changing rooms, and my dad got offered 'extras' for extra money, followed by demand for extra money not to provide the extra services!
Highlight of last trip was the ceramic tiled mosque over on the Asian side - it's 'top 10' lists of things to see, but you need to get the ferry over and then a cab up the steep hills, but thoroughly worth it. The imam showed us all round and was delighted to see us as apparently they only get a couple visitors a month. Then got ushered into the adjacent hall with a little tea room (just like a church hall) and treated to tea. Wandered back to the ferry through the residential areas - much, much poorer than the European side. But again some excellent cheap restaurants.
If there's something expensive you want to buy, do set aside a couple hours to haggle over it and enter into the spirit of it. We got leather coats, MrNC's made for him, collected it a couple days later. Only irritant is people trying to sell you carpets all the time, but my friend from Istanbul says it happens to locals too.