Both airports are fine, really. International flights used to go to Ataturk (the European one) slightly more often, but I've flown out of the asian one a few times too. if you have a choice and not too many bags, Ataturk is connected to the metro line, so you can get to Sultanahmet or Beyoglu (where most hostels are located) with only one change. To travel from the Asian airport, you'd have to take two trams, a ferry, and maybe the funicular.
Still, if you don't want to navigate the metro/tram system by yourself, both airports have shuttle busses to Kadikoy and Taksim leaving every half an hour. They are called Havas (Havash) and they are super convenient. The one from Ataturk costs around 10tl, while the one from the asian airport is I think 14tl. They'll also accept Euros or Dollars, but won't give you a great exchange rate.