The standard approach to the Asian side is by ferry from the European shore. The İDO and Şehir Hatları lines connect Eminönü and Karaköy (near Galata Bridge) to Kadıköy and Üsküdar frequently throughout the day — the crossing takes 20–25 minutes and costs the same as a metro token on an Istanbulkart.
The Marmaray rail line crosses under the Bosphorus and connects Sirkeci (European side) to Ayrılıkçeşme (Asian side) in minutes — useful if you're heading further into the Asian suburbs or connecting to the metro network. The metro M4 line runs from Kadıköy south to the airport.
By car, the Bosphorus Bridge (15 Temmuz Şehitler Köprüsü) and the Fatih Sultan Mehmet Bridge carry road traffic between the continents, but traffic during peak hours can be very heavy.