There are so many places to see in Istanbul, and trust me, it will take a while to see every neighborhood in the city.
Depending on the lenght of your stay, you can also choose to stay at two different places. I know that this strategy works for many people who are pretty mobile, and don't carry a ton of luggages around. If this is your first time to Istanbul, then you'll probably be spending the first two-three days in the old city, Sultanahmet area, checking out all of the historical landmarks. Once you're done with the touristy stuff, you'll want to hang out where the locals live and breath. And, that's definitely not going to be the old city. If you are not staying around Sultanahmet, then I suggest you to find a place, which is centrally located and you can get to wherever you want ASAP.
My personal favorite is Galata neighborhood. It has many expats living there, and always crowded with travelers & happy faces. You are very close to Istiklal Avenue, and can take the metro from Sishane to go to the north side (Sisli, Nisantasi, Etiler, Bebek) of Istanbul conveniently. Also, you can just walk down to Karakoy to take the ferry to Kadikoy (asian side) or walk over the Galata Bridge towards the old city. It's very centrally located.
In this map/picture, Galata Tower is marked as No:1. Bottom of the map is the old city direction (south). Here is a picture of Galata neighborhood, and its old houses.
Choose to stay near Taksim Square, if you want to be close to nightlife and other local hangout places. For more touristic attractions, Sultan Ahmet area has quiet a few boutique hotels in its side streets. Hotels for all budgets can be found in this area, including a youth hostel. If you would like to be more secluded from the city's chaos you probably can find some boutique hotels (and a few 3-4 star hotels) or airbnb hosts in some of the neighbourhoods along bosphorus (on Anatolia side Cengelkoy, Anadolu Hisari, Uskudar ... and on European side Rumeli Hisari, Bebek, Ortakoy).
Galata, Cihangir, Ortakoy, and if you can afford them, Nisantasi and Bebek are some of the nicest neighborhoods in the new city. Galata and Cihangir are close to Taksim, while the others are in their own little city centers. I'd suggest avoiding the old city area unless you are ONLY interested in history/carpet buying, because it's kind of a slog away from all the action, cuisine, and nightlife in the new area.
Resources:
Get Cultured: Istanbul – A Guide to this Eclectic City and its Hip Neighborhoods
Istanbul: Best neighborhoods to stay in Istanbul
