—> The best hostel I stayed at was the Taz-mania hostel in the Beşiktaş area. You can take a HAVATAŞ into the city from the airport (to Taksim) then a taxi to Beşiktaş. The Beşiktaş area was great. Be sure to have lots of chai and smoke from a hookah at a street cafe. Craft beer can be found at Taps Brewery in Bebek if you find yourself craving it (I did-there is a very limited selection of beer/wine around!!).
My favorite recommendation for you: Take a ferry to the Princes Islands (aka Adalar)! I thoroughly enjoyed a couple of days exploring the small island (the first ferry stop). There are no cars, many little paths through the vegetation lead to Mediterranean beaches. I left from the Besiktas ferry dock next to Bahçeşehir Üniversitesi. http://www.turkeytravelplanner.com/go/Istanbul/Sights/adalar/trans.html
—> From Beşiktaş, you can also walk to Dolmabahçe Palace easily. The Galata Tower is nearby also.
—> Public transportation is pretty good (the train). That will take you to the Grand Bazar (aka Kapali Carsi). We went in the evening which seemed like a good time) and the Sultanahmet district tourist is full of a day’s attractions. Go to the Hagia Sofia and Blue Mosque then sneak behind them for a tea in the neighborhood (walk towards the 4 Seasons)
For nightlife, head to Taksim. Note that you will be out until well into the morning!
—> A ferry trip to Maiden’s Tower (Kız Kulesi) will bring you to the Asia side of the city as well. This tower is in the middle of the Bosphorus.
—> One thing that I was not able to do but am DYING to return for was a visit to Cappadocia. I recommend googling: “Cappadocia air balloons” and let me know when you’re going—I’m coming!