Just spent 5 days there! Blue Mosque is free, just go during non-prayer hours and dress respectfully.
Hagia Sofia, Topkapi Museum & Harem are a must for most people. There is a 72 hour Museum Card that will give you front of line access to those sites and a few others and discounts on the gift shops too. There are other museums included with the card as well to make it better value. It was 72 TL for the card. We also saw the Chora Museum, but there are about 4 others.
Get an Istanbulkart from a shop next to the train or tram lines. You will save lots of money on the transit options in the city and you get 6 TL back when you return the card.
If interested in a hamam, Turkish bath, go to the Asian side. The European side charges major tourist rates like 30 euros or more each. We went to Cinili Hamam, which is from 1640, and paid 30 euros for two people for the bath with massage. about 7 years ago
Main square in european side is named taksim and main street called "istiklal" starts from taksim, the region is also named as " beyoglu". Main side in asian side is called " kadıkoy"
in europe side there are nice cafes in "ortakoy" easily reacheable from taksim by taxi for about max 10€ almost 7 years ago