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.

I agreeI disagree

1.  
2. Send to friends

Beware - especially if you're a single male - of being approached by seemingly friendly blokes who start chatting (usually by asking you the time and then saying 'Oh, sorry, I thought you were Turkish...'), invite you to drink with them, particularly around Istiklal Caddesi. This can lead to you being relieved of your money in a dodgy nightclub. It very nearly happened to me.

Istiklal Caddesi

I agreeI disagree

1.  
2. Send to friends