Let me know if you have any questions.
#1 Listing and Selling Agent
Keller Williams West LA
These are pleasant walkable areas, and you can still get downtown easily in about 15 minutes.
I would advice against living downtown. There are some problem areas and it's not the safest place for you.
Good luck in your search.
I'm actually looking for a place to rent - a studio or one-bedroom at most - hopefully for under $800/month. I've seen a couple of places so far, and though the neighborhoods seemed perfectly safe I have heard quite a lot from friends and coworkers about finding a safe neighborhood in LB. (Their opinions seem to be that there aren't many, but I don't think that's true - I just don't know where they are. Or, rather, which areas to completely avoid.)