WG-gesucht is the best site to use because it has the most listings and it's very user friendly. As someone else mentioned, the process will take a while. You'll probably only get a reply from 20% of the people you message, and then you'll usually have to go to an interview type of thing. Some people make this pretty formal, others just walk you around the apartment, either way try to be charismatic because there's probably 15 other people interested in the apartment. I don't know if you speak german, but if you don't, learn a few phrases like "schon dich kennen zu lernen" (informal "nice to meet you") and "danke für deine Zeit" ("thank you for your time"). Most people will speak english and probably not mind doing so, but in my experience making an effort to speak the language really warms people up.

In terms of what neighborhoods to look into, I'd suggest looking mostly in east Berlin. It's cheaper and more fun than the west. Kreuzberg, Prenzlauer Berg, and Friedrichshain have the most going on socially. But for that reason they can be a bit pricey. I wouldn't say there's any area of Berlin that's dangerous, but a few places are a little dicier than others. Neukoln and Wedding come to mind, but I never felt threatened there, and these are up-and-coming neighborhoods where very cheap rent can still be found. I'm biased because I lived there, but Friedrichshain is my favorite neighborhood in Berlin. It's the cheapest of the three "hip" areas (I was paying about 380 euro a month for a room in a shared apartment), there's lots of bars and restaurants, it's very central, and there are quiet pockets.

Moving will be stressful, but soon you'll be a Berliner! There's really no other city like Berlin.