Learn the layout of Los Angeles!
Dorothy Parker once said, "L.A. is 72 suburbs in search of a city", and she is absolutely right. Los Angeles is a sprawling sea of multi-cultured neighborhoods and landscapes. It's important you get a good idea of the different neighborhoods before you start looking for apartments. A listing of neighborhoods can be found here http://en.wikipedia.org/wiki/List_of_districts_and_neighborhoods_of_Los_Angeles and map here http://www.losangelesmap.org/map-los-angeles.html
In a crude breakdown one should know that Northern L.A. is typically for families, East L.A. has it's share of crime but is slowly becoming gentrified, West L.A. is nice but expensive since it's near the water, Central L.A. is affordable but congested, and South L.A. typically should be avoided.
Talk to people!
Contrary to popular belief, Angelenos are typically nice people. Don't be afraid to ask questions. Los Angeles is full of people who came to the city fresh off the bus just like you! Residents can tell you what areas to check out and which areas to avoid. I'd recommend going to a cafe in Hollywood and asking around.
Drive around first!
I've found that driving around L.A. looking for an apartment is your best bet. You will see A LOT of "For Rent" signs year round. Trust me. Keep a notebook with you, write down the phone numbers on the signs, and start calling away. Once I found an apartment right across the street from where I worked. I was the first person to call and got the apartment that day! Make sure you bring your Thomas Guide or GPS system.
Check out Craigslist and Rentslicer!
Craigslist is a great way to see what's out there. However, I've noticed in my experiences that mostly expensive listings are on there. Craigslist is a great resource and starting point, but don't settle for the first place you see on.
Another interesting resource is www.rentslicer.com. Rent Slicer hunts through the internet for apartment listings in an area, then averages the prices. You can type in any zip code in the country to find the average cost of a two bedroom apartment. However, the average is higher than in actuality. Just another good reference point.
Check out the LAPD's crime map!
Another helpful resource is the LAPD crime map. This website will list all crimes that have happened in the area you type in. Don't be shocked if a lot of bad stuff has gone down in your neighborhood. It's L.A. remember. The home of the Watts riots, the Manson murders, and Jack Nicholson's golf club whacking.
www.lapdcrimemaps.org