Turkish Kebap
We have many different types and it's a bit different than what "kebab" is in Europe. The classic would be Adana Kebap (spicy) or Urfa Kebap (not spicy Adana Kebap). In a good restaurant, I recommend a "plate." In more of a casual spot, you can get it in dürüm (wrapped) version.
Döner
Same as kebap, slightly different than what is döner in Europe. In a good restaurant, I highly recommend a plate or even Iskender (döner plate with special tomato-sauce and butter). In casual spots, you can get a wrap or sandwich versions as well.
Turkish Delight
Ali Muhiddin Hacibekir (Best and one of the only traditional Turkish delight shops, family-owned and operated since the year 1777). They have locations in Eminönü, Taksim, Kadıköy. I highly recommend not buying Turkish delight anywhere else, especially grand bazaar or Istiklal Street.
Lahmacun & Pide
Turkish style meat-pie and Turkish style flatbread.
Seafood
Turkey have a massive list of seafood available. I highly recommend going to a nice restaurant, with a nice view just by the sea, and trying different fish. Whatever is in season, go for it as a big fish. It's always good to support it with small mezes (appetizer plates which they'll have many of them) & calamari.
Desserts
Baklava (Try in Güllüoğlu, one of the few traditional ones left in touristic areas: classic one is the pistachio one), Künefe, Sütlaç.
For Foodies
Streetfood of Istanbul: Börek (Flaky pastry with cheese, spinach, or meat), Kokorec (intestines), Islak hamburger (wet burger), balık ekmek (fish sandwich), midye dolma (stuffed mussels), Kumpir (baked potato with toppings).
Regional Kebap Types: Some that are more of regional delights that you might come across in Istanbul: Cağ Kebabı, Testi Kebabı, Tas Kebabı.
Home Cooking: Some of the food that's prepared in most Turkish houses: Mantı, Sarma, İçli Köfte, Karnıyarık, Hünkar Beğendi, Dolma, Kuru fasulye.