As person who was born, raised and grew in Istanbul, I did travel almost every inch of Istanbul due to my jobs. . I see tourists everyday in Sultanahmet area and its really packed with tourists and its going to be packed with tourists more and more in coming months.I will go back to Canada soon ( I have been to Turkey for last 8 months)
Probably all of you who want to come to Istanbul will see Grand Bazaar, Blue Mosque, Hagia Sophia etc..
I would like to give you some of my recommendations worth to be visited in Istanbul and they are either non-touristic places or they are known by less tourists
Feshane: I do recommend this place to visit. There are 81 cities in Turkey. Every city organises 3 or 4 days exhibition in Feshane almost every week. They run from Thursday to Sunday evening or from Friday to Sunday evening depends on city.
Cities exhibit their local products, their arts, culture, cuisine, traditions etc.. Its like a fair. Every region has different culture, cuisine, music and customs in Turkey so it is really worth to visit Feshane.
https://www.youtube.com/watch?v=RcJyqbrC_7g
https://www.youtube.com/watch?v=iZi-IFKUp5M
Two links above could give you some idea even if its in Turkish.
http://www.feshane.com.tr/tr/etkinlikler1.html
The link above shows time table and its in Turkish. They always update timetable.
Its not that far from Sultanahmet. Taxi will cost you about no more than 15 TL ($6USD) from Sultanahmet area. You can take bus from Eminonu too. Or if you like walking , you can walk by Golden Horn and visit Greek Orthodox Church in Balat on the way to Feshane.Ortakoy, it is worth to visit to eat 'Kumpir' and watch Boshborus Channel.
Bebek: Its really worth to visit Bebek and walk from Bebek to RumeliHisari by watching Boshborus. You will see a lot of high end cars on your walk, luxuorus yatches, high end fish restaurants etc.. Dont forget drinking coffee at Starbucks in Bebek. You may not like coffee but it is worth to sit down and have a drink by watching Boshborus.
Besiktas : Worth to be visited because of its night life, bars, restaurants, cafes etc....
5:Uskudar: Take the ferrry from Sirkeci and see Uskudar in Asian side. Its beautiful place to be visited.
There are more places I would recommend to see most. They are close to Sultanahmet Area and they are convenient to be reached by transportation or by walk.
In addition to these places, If you have time, I would recommend to see 'Ayvalik' in other side of Marmara Sea but it will take about 6 hours to go to Ayvalik with ferry and bus after ferry.
If you go to 'Ayvalik', you can also take ferry to Greek Island Mytilini from Ayvalik. Ayvalik has Sarimsakli Beach too. Its very good spot to swim. Its famous for its Aegean cuisine( They mostly use olive oil in most of their cuisine)
Two google link can give you some idea about Ayvalik and Mytilini:
https://www.google.com.tr/search?q=ayvalik&es_sm=91&biw=1440&bih=689&source=lnms&tbm=isch&sa=X&ei=P5NOVdqAE4r1UofTgPgM&ved=0CAYQ_AUoAQ
You can also take flight to go to Ayvalik from Istanbul to Balikesir Edremit Airport.
Good luck