As long as you don't have loads of bags and it's not the middle of rush hour, it's perfectly easy to get the train/tram into (and from) town. Take the LRT train from the airport to Zeytinburnu, then change to the tram that will take you all the way to Sultanahmet and beyond. You can also change at Aksaray, but I recommend Zeytinburnu (even though it will take a bit longer) because it's an easier interchange (just over a bridge) and you're getting onto the tram at the start of its journey (so you're more likely to have space to get on), plus there's an Akbil kiosk if you want to buy one of those multiple-trip gadgets. Otherwise, the trip costs 2.20 YTL for two tokens.
Airport, Zeytinburnu.
There are 2 airports in Istanbul, Sabiha Gokcen is on the Asian side. There is a bus from outside the terminal to Taksim Square for about £3.50 single (takes an hour or so). Alternatively you can take a taxi to the local station, then a train to Kadikoy and then a ferry to Eminonu. Longer, more expensive (no more than £5) but arriving in Istanbul by ferry is definitely worth the effort.
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