I would recommend taksim square over Sultanahmet. Sultanahmet maybe close to the main touristy sights of grand bazaar, blue mosque, topkpai palace and Aya sofiya but there is a whole load more to Istanbul then Sultanahmet and it would be good to get away from the tourist hordes and see some real istanbul life. taksim is the central spot and on the end of a metro line to go to other places other then sultanahmet.
IE, Dolmabahce palace, Ortakoy, Bebek, Kadikoy for sali pazari, princes islands, golf at the KCC, historical district of Arnavutkoy, Rumeli hisari and if feeling ventouros goto Anadolu kavak where not many tourists go but is stunning.
Sultanahmet is not everyday Istanbul and as a resident here for more years then I care to count, I maybe goto that area about once a year if that just like many other Turks do.
Still a great place to go if you have never been there but try and get a feel for the old and new. You have Sultanahmet for all its history but you will be surprised when you drive up buyukdere caddesi to Maslak and see how modern the place is getting with the current tallest skyscraper in Europe the istanbul Sapphire, the beaches at Kilyos......too much to mention.
