I hope you got a job lined up already. Aspiring TV writers are a dime a dozen here.
But seriously, if you wanna get into the entertainment business, I think you should move to Culver City or North Hollywood or Burbank. Sony Studios is in Culver City, along with a handful of independent studios and production companies. Culver City is kinda boring to me. But the location is good.. Santa Monica is to the west, downtown LA to the east, and the Valley to the north. About 30 minutes in each direction, depending on traffic.
There are lots of cable and broadcast TV studios located in Burbank. Burbank is nice and clean, but kinda boring. Occasionally you'll spot celebrities or porn actors at the market. Other than that, it's a pretty sleepy city with affordable housing.
North Hollywood is considered the arts district for whatever reason. Some parts are good, and some parts are bad. There's a Metro Rail station there, so you can hop on the train to Hollywood, downtown LA, Pasadena, Long Beach, and Compton.
I hope you're bringing a car or planning on buying one when you get here. Our public transportation kinda sucks and everything is so spread out, you need need a car to get around. After paying rent, a chunk of your paycheck will go towards paying for gas from all the driving you'll be doing.
Check Craigslist for apartment rentals. You should be able to find a 3-bdr apartment for $2000. Or maybe even a small house.