While it will sound crazy to anyone living here, you can get by in LA without a car.
It depends where you live and what you do. While Los Angeles is a car-centric city, my wife lived here for two years before she finally got a car. She lived in Hollywood and she would walk or take the bus to work and take Taxis at night. Los Angeles does have a decent bus system, but the city is sooo huge that it can be difficult to get where you want. In addition, there is a metro rail system. Unfortunately, it does not extend to the Westside of LA. They are currently building the Expo line which will ultimately span from downtown to the Santa Monica pier, but that should take another ten years.
While we Angelino's could never live without our cars, a New Yorker like yourself could likely manage without one, with a few stipulations:
1- You have to live close to work or next to a bus line that takes you directly to work.
2- You have to be OK spending a good portion of your income on taxis if you like the nightlife.
3- You have to have friends living in LA that you can rely on for those time when you need a car.
Your bigger concern should be the withdrawl you will go through from not being able to get good pizza!