Whilst it is possible to make that trip using some combination of Metro/Public Bus/Ferry, the best option is instead to catch one of the airport buses run by either Havabus or Havaist. These leave SAW roughly every 30 minutes, 24 hours a day (Havaist only for the 24 hours/day), and have dedicated luggage storage under the bus.
These buses will take you to either Taksim (Europe side) or Kadıköy (Asian side). Which of those destinations is the best option will depend on exactly where you're headed on the Europe side, however given the luggage you'll have the best option is likely to head to Taksim and then take a taxi (or Metro) from there to your accommodation. Alternatively you can catch a ferry from Kadıköy to somewhere near your accommodation (eg, Eminönü, Karaköy, Kabataş or Beşiktaş) and catch a taxi/Metro from there.
Whilst not the outright cheapest option, it's far cheaper than a taxi, and the best compromise between price, convenience and time - especially given you are carrying luggage.