Istanbul is a big city and people get confuse where to stay. Wherever you stay you see a different side of the city so if you think like this, it stops worries.
There are mainly 3 districts
Sultanahmet is a touristic area. Everywhere tourists with their cameras and locals who try to sell something. But have many hostels and hotels, close to tramline.
This place is full of life, crowded and has many hotels and hostels. Night life is busy.
Kadıköy is a downtown on asian side and quieter than Beyoğlu. But the real Istanbul is on the European side. For a longer stay, Kadıköy is good but for shorter stays Sultanahmet and Beyoğlu is better.
Wherever you stay check the transportation options!
You can stay in a local’s house by http://www.airbnb.com
Or find a hotel or hostel from http://www.booking.com