Hi! Area called Sultanahmet is full of interesting and old places. Aya Sofia, Ahmediye (Blue Mosque), Topkapi palace, monuments, also Harem. It's all in the European part. Don't buy things on Kapali Charshi (COvered Bazaar).
THen the centre of asian part - Kadikoy. Stores.
What else. Haven't been by myself, but have heard about Princes Island, Umraniye.
Also visit site www.istanbul.com.tr
If you can read in Russian, then it is worth reading on turkey.ru about Impressions on Turkey. THere is interesting story about visiting Istanbul on May.

Snezha