Turkey's YHT High-SpeedTrains from Ankara, Eskişehir and Konya arrive at Istanbul's Pendik YHT Gar (High-Speed Train Station) on the Asian side of the Bosphorus near Sabiha Gökçen Airport (map). Previously, the first stop in Istanbul for the high-speed train was in Pendik, but now the trains come up to Söğütlüçeşme. There is also a Bostancı line between Söğütlüçeşme and Pendik.
In March 2019, the Marmaray suburban train line was opened to operations between Halkalı on the west side of Istanbul (the station for trains to Europe), to Gebze to the east, making travel between Pendik and both shores of the Bosphorus fast, easy and inexpensive.
Obviously, the easiest way to travel from one transport terminus or hotel to another is by private transfer, a car or van reserved for you and your travel companions. It's more expensive than public transport if you're traveling alone, but more comfortable and secure, and it may be faster. The larger your travel group, the more a private transfer makes economic sense.
Companies that do airport and port transfers can also take you to the Pendik YHT Gar.
Pendik YHT Gar to Kadıköy
Take the Marmaray toward Halkalı to Ayrılık Çeşmesi, then the Metro (M4 line) to Kadıköy.
By City Bus
Walk from the Pendik YHT Gar south along Gazipaşa Caddesi (map) toward the sea and the İDO İskelesi (İDO Ferry Dock: ask anyone) to the İETT bus stop at the ferry dock. E-9 buses traveling from left (east) to right (west) will take you to Kadıköy, from where you can take a ferry to Eminönü near the center of Old Istanbul.
The other buses which you can use to reach the Pendik YHT Station are 132K, 133T, 16D, 252, KM13. You can check this website for more options.
C-88 minibüs (dolmuş) goes to Pendik Station as well.
Pendik YHT Gar to Old Istanbul
Pendik YHT Gar to Beyoğlu
Take the Marmaray toward Halkalı to Sirkeci, then the Kabataş Tram to Karaköy and the Tünel to Tünel Square or take the tram to Kabataş and the Füniküler to Taksim Square.
By Ferry & Tram
Follow the instructions above to reach Kadıköy, then take a ferry to Karaköy and take the Tünel up to İstiklal Caddesi, or the Bağcılar-Kabataş tram toward Kabataş, then the Füniküler (fenicular) to reach Taksim Square.
An alternative, probably faster and certainly more comfortable (though you do miss the cross-Bosphorus ferry voyage), is to follow the instructions below to go to Sabiha Gökçen Airport, then take the Havataş Airport Bus to Taksim Square.
From Sabiha Gökçen Airport
IETT city buses depart from outside the Arrivals level of the airport terminal. Buy tickets or a pass from the kiosk near the buses.
IETT Bus E-9: Take the E-9 bus and get out at the Pendik İDO (PEHN-deek EE-doh) stop, the one for the İDO ferryboats to Yalova. The ride takes about 20 minutes.
From the Pendik İDO bus stop, walk inland through the parking lots and to the right of the small Police Station, then continue inland along Gazipaşa Caddesi to the Pendik YHT Gar, a 10-minute walk.
IETT Bus KM20: The KM20 bus from Sabiha Gökçen Airport also takes you near the Pendik YHT Station, but it runs only every 50 minutes (9 buses daily). The YHT Gar stop for the KM20 bus is on Abdülhak Renda Caddesi on the north side of the station (map).
IETT Bus 132H: The 132H bus from Sabiha Gökçen Airport takes you to Pendik YHT Station as well. Check out the departure hours from here.