It's a fantastic city, one of the most amazing places I have ever visited. The people are lovely, among the most friendly you will ever meet.
I agree with Slapsie that one should stay in Sultanahmet, then all the biggest attractions are within walking distance. You can cab it to Taksim (or take a tram) and get the ferries if you want to see the Asian side.
Be sure to have lunch at the Sultanahmet Koftecisi, it's a famous old place where they only have a few items on the menu, but they are really great. Right on the main tourist drag, but still a locals experience (though there'll be lots of tourists too obviously). There are several restaurants on the same street with basically identical names, so check you've got the original. See http://www.istanbult...hmet-koftecesi/
The Orient Express restaurant on the main Sirkeci railway station is a fun experience. Food maybe not top drawer, but a beautiful room. Lucy and I ate dinner there on the night of our tenth wedding anniversary, after watching sunset from Galata Bridge (not quite as romantic as it sounds, six lanes of traffic, two of trains will do that for you) but still cool.
We stayed at Sirkeci Konak (http://www.sirkecikonak.com/) at the recommendation both of Trip Advisor and a contact who does PR work for the Turkish tourist board, and it's lovely. Right below the walls of the Topkapi palace, easy walk anywhere in Sultanahmet, independently owned, wonderful welcome. They have a good restaurant too. It was around 200 euros I think, at the top end of what I'll spend, but definitely worth it.
The Grand Palace Mosaic Museum in the small bazaar below the Blue Mosque is a don't miss. They built the mosque on top of the old Byzantine palace, and archaeologists have dug up a bunch of amazing mosaics. They're in this crappy little museum tended by one guy and a couple of stray cats, and any one of them would be a star exhibit in a major European city.