Travel to Istanbul

Hi, My name is Anita and I am needing advice about airport transfers from the Asian side of Istanbul from Sabiha airport to Taksim on the European side. Air Baltic lands at Sabiha and I will be staying in Istanbul for 3 days before leaving from Ataturk airport on the European side. I will have a 30kilo suitcase plus backpack and minimal funds so I want an airport bus to get me to Taksim rather than a taxi. I read on the Internet that there is a Havatas Coach that takes passengers from the airport to Taksim. Has anyone any suggestions what the simplest way to get from Sabihah airport to the centre of Istanbul would be and where to stay on a budget. I have not yet booked my hostel.
Anita