Hi Soheir! I've been to Turkey but not necessarily for bellydancing. I visited a good friend of mine there. As for bellydance performances you should be careful because, as far as I know, there are lots of places for "tourists" and the quality is not perhaps the best. You could definitely do some lessons. Check out Ahmed Ogren's website.
http://www.turkishgypsydance.com/eng.asp
He's a great Turkish Rom dancer and he might be able to recommend some places in Istanbul to see some good dancing. You could also send an email to Artemis in DC...she might be around to make some recommendations.
As for costumes, if you find yourself in the Grand Bazaar you'll see lots of stalls with cheap "tourist" bellydance costumes. Tell the people there that you are looking for professional costumes and they'll point you in the right direction. You could also go to *Shira*'s website for some recommendations. Of course, find out where Bella, Marta the Legend or Sim Moda is and go there, too!
Sultanahmet is the old city where you'll see all the old sites (Blue Mosque, Aya Sophia, Topkapi Palace, Grand Bazaar) but also check the Egyptian Spice Bazaar (in Eminonu) and the Taksim district (young, hip, cafe-culture). A great website for help is:
http://www.turkeytravelplanner.com/index.html
The guy who runs this site is a writer for Lonely Planet and Frommer's Guides.
Good luck! Istanbul is great...lots to do and see!