Slap up meal, for a special occasion: there is an amazing fish place called Esmet Baba, I think the area it's in is called Kuzbuncu. Since you don't like arak/raki/ozu, drink the Efes.
When drunk: Get an islik burger. It's a burger which has been marinated in some kind of liquidy ketchup. You can find them all over Taksim Square, but only at night.
On the Island, Bayukada: Bizarrely enough, some Greek monks run a kebab restaurant at the very top of the place. Very spartan, but the food is better than anything in Haringey, I promise. Well, it's all the same, but the quality is better.
Haci Abdullah Kebabs: in the European side. Yes, I know the kebab thing is getting old, but ....
Also, if you walk from Taksim Square to Galata, you will find an unmissable little alleyway where all the little coffee shops play live music. It's kind of amateurish, but fun; mostly beer and drinks here and not food, but worth a stop.
For something Western: try the Istanbul Culinary Institute. It's Zagat rated, but not too pricey.
For Turkish breakfasts: Walk along Istiklal, which is the big boulevard off of Taksim, and find these little shops with white tiled floors and walls. They're unmissable. The food is basically cheese, spinach and potato pastries. Again, the same type of thing as you get in Haringey, but here they use more grease.
For goat's milk ice cream (I promise you it's good): Mado, which is on Istiklal Cadessi.
Cayih: It's an area called Kadikoy. It's 3 restaurants in one, and is a good place to escape the tyranny of the kebab.
Tripe Sandwiches: there is a whole arcade of them just parallel to Istiklal, around the corner from Taksim. I was deceived by a friend into biting into one, but it was well worth it. Kokerec it's called in Turkish.
Sardine sandwiches: there is something called balik ekmek ("fish-bread"). These are found at piers when you wait for a ferry, say to the Bayukada Island. They cost nothing and they're tasty. The nicest one I found was near Emminonu, another district.
So the thing about Istanbul: they like they're food; it's really well priced, really, really well priced; and it's very clean, so you're not likely to get food poisoning. You could ignore everything above, take a random walk with a few Euros in your pocket, and you would have had some amazing meals. I've only ever had one bad meal, and that was a place near the Sultan Ahmet Mosque which was clearly meant for tourists, so both Martin and I should have known better.