Good call on deciding to rent an apartment. It is the best way to have a local feel of Istanbul. As far as I know, people who are renting their houses are really friendly and hospitable. They help you out with itineraries, local information, etc. Galata neighborhood would be my first suggestion. There are so many places to see around Galata, and great vibe. Cihangir would be the second suggestion considering the fact that most expats live in that neighborhood. You should check www.airbnb.com for rentals. Many locals rent their houses through that website. Also, you can read reviews about the owner of the house. Anywhere close to Istiklal Avenue would also be safe and close to public transportation, good food, local life, and nightlife.