I live here. Have for eight years.
It's safe as far as American cities go. I've never had a crime committed against me. I have always lived on the westside, which is considered generally safe.
There's no reason a tourist would even want to go to the eastside that I can think of, and that's where all of the historic and stereotypical crime you've mentioned happens.
By far, the worst thing about LA is the traffic. It's one of the most poorly designed cities in the country. Driving is an incredible pain and the natives have learned how to deal with it by driving aggressively. Avoid driving long distances that require using the highway during rush hour.
Anyway, it's a fun city to visit, so you'll probably enjoy it. Good luck!
Also... the water is far warmer than it is in New Jersey, but colder than it is in Florida. At this time of year, I don't advise going to the beach for the water anyway unless you're into surfing.