I'm from Istanbul but I've moved to London last year...Anyway, you should go and see the Bosphorus of course ( you could go on a tourist ferry). You should see Haghia Sophia, Topkapi Palace, Blue Mosque, Grand Bazaar( keep an eye out for jewellery), Yerebatan Sarayi which is near Haghia Sophia if I can remember correctly and is an underground cistern ( btw it's where they shot a scene of a James Bond movie ( From Russia with Love
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You can go to Istiklal Street in Beyoglu which became really popular in the last few years (it used to be considered too 'hippie'),there are some very nice cafes and bars and shops. You could go to Dolmabahce Palace and also to the street market which takes place on Sundays in Ortakoy which is along the Bosphorus and is something between Portobello Road and Covent Garden in London. (sorry about the comparisons
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You can go to Nisantasi, but I wouldnt say it's a very touristic place, you would see lots of women with dyed blonde hair and Hermes bags and shops like Valentino, Louis Vuitton etc. ( I dont think you would want to do designer shopping in Istanbul, they are much more expensive over there...) You could go to Beymen though, which is a department store and again sells designer clothes for twice the price but also has its own label. If you go Nisantasi, you should go to House Cafe, they do the best salads but as I said, it's not a touristic place, it's just where fashionable Turks go to...
For food, you should go to the Tarihi Sultanahmet Koftecisi to eat meatballs( koftes) which is very cheap and it's near all the major sights in Sultanahmet. For the best kebab, go to Beyti, Kosebasi and Tike. For spicy south-eastern food( Arab-Kurdish) go to Hamdi in Eminonu.(The food there is just delicious, believe me). If you want to eat fish with meze( appetisers) which tourists call Bosphorus meal, then you can go to Kiyi Restaurant in Tarabya which has very good views of the sea.
That's it ,I guess...
If you have any other questions, you can always ask me..