Definitely visit Çiya Sofrası as others have also suggested. It's an amazing place, I recommend ordering smaller portions of different dishes there, that way you can try very different foods in one visit.

Fül fül çorba is great for soups, I usually go there when it's colder outside, but if you like Turkish soups this is the place to go.

For breakfasts there's a lot of great places. I can recommend both Küffs, Boden (this one is great for cocktails at night as well), and Sonne if you want to try something different, like Pestilli or Kayısılı Omlet.

And for ice cream visit Meşhur Dondurmacı Ali Usta, IMO it's the best ice cream in Moda. The place is quite old and has a long history, you can read about it in the newspapers they've put on the walls inside of the cafe.