Can anyone help with planning a trip to LA?

Hi, I am hoping to visit LA for the first time in May or so of this year. I'd like to see Beverly Hills, Rodeo Drive and Hollywood. Also the Walk of Fame, Santa Monica Pier, and maybe the Griffith Observatory. What else is good to see? Also what part of town is best to stay in for the things I'm interested in? I am on a tight budget, so I would need an area with at least a couple of cheaper hotel options. Is it safe in LA for a solo female traveler? Are there any good consignment stores in the Beverly Hills/Rodeo Drive area? (Lord knows I can't afford anything in any of the other stores...) Is there any chance of coming there and not having to rent a car? How is the public transportation? Which restaurants are recommended for the budget minded? (I definitely want to go to In-N-Out Burger as I've never been there...) Thanks for any help you can give!