After feeling a little let down by the Pera Museum I was genuinely taken aback by how much I liked the collection at Istanbul Modern. Spread over five floors, its exhibitions (both permanent and temporary) showcase contemporary art that's both obviously good and thought-provoking. Modern stuff often leaves me cold or loses me for its pretentiousness, but I really did appreciate a lot of the pieces here.
As you would hope, there are a lot of works by Turkish artists (the photographic exhibition of Ara Güler on the top floor at the moment is outstanding; also keep an eye open for Ramazan Bayrakoğlu's 'Fire' and Sabire Susuz's 'Shopping', both extraordinary for their detail) but also quite a few by international names (highlights for me including Zhan Wang's 'Artificial Rock' and Olafur Eliasson's 'Red Emotional Globe'), all of which sit very well together.
Tickets are Ł60 or €9, but while the price is on the steeper side by local standards it's well worth the extra investment. There's also a nice little café - which is a must for any museum worth its salt - where the prices are more (than) reasonable, so it all balances out. Easiest access if you're coming from Sultanahmet/Fatih is via tram and then the funicular up from Karaköy, known as Tünel.