I'm going to visit Istanbul in April, an I'm curious about the best way to get to the Taksim Square from Sabiha Gokcen International Airport (SAW) by public transport.
The shuttles are white colored and parked in front of the terminal building arrivals level exit ready to take you to Taksim for a fee of 14 TL and to Kadıköy for a fee of 9 TL. Shuttle’s Taksim departure point is across Taksim Divan hotel, in front of the Point Hotel.
As mentioned below Havatas looks the best way to go Taksim which is cheap and one-way drive. However considering the peak hours, it may take up to 2 hours to go to Taksim with Havatas. So it will be better to check the peak hours and may spend time at the airport for a while.
If you have internet connection you can check the traffic jams on the Bosphorus Bridge via ibbtrafik app. (It is government's app and I do not aim to advertise :)
Here's the rustic way with more transfers, though under certain traffic conditions it can be quicker. I'm partial to the very scenic ferry, funicular and tram rides and I travel light.
You can also take bus from Göcken Airport to Kadıköy (9 TL), approximately 60 minutes.
In Kadıköy get a fare card if you plan to use public transportation in Istanbul during your stay (6 TL for the card + load fare for the following trips) and transfer to the ferry to Karaköy (2 TL), 20 minutes.
From Karaköy you could just walk up the hill past the Galata and through Istikal (main pedestrian shopping zone) 30 min. Or take the Tünel underground funicular to Beyoğlu (3? TL), 2 mi. From there take the tram (3? TL), 10 minutes down Istikal to Taksim.