It depends on what you want to do. In general, the best historical things are on the European side- although there are a fair share of palaces and whatnot in Asia, they aren't nearly as well connected to public transportation. Your best bets in Asia are Kadikoy, Moda, and Uskudar.
Uskudar is historically the closest neighborhood to Mecca, and thus has some of the oldest and most beautiful mosques in the city. It's a good way to see some of the more pious life-styles in the city.
Kadikoy, on the other hand, is a great place to go for dinner and drinks. It's full of fun restaurants (Try Ciya!) and is a good place for cheap fish sandwiches and the like. The city's biggest collection of "rock bars" is to be found here, on the aply named Barlar street.
Moda is a great and relaxed neighborhood past Kadikoy towards the sea. It's got a fantastic park, tons of cafes, and a wonderful collection of upscale turks and their beautiful pet dogs.
