If you want to see things while you are there, you need to rent a car. As was said, public transit is awful in L.A.. The place is also a sprawling massive city, and cabbing would be far more money than renting a car.
As far as places to stay, Santa Monica is my favourite. It is right on the beach with the Santa Monica Pier right there. From Santa Monica, you can walk down to Venice Beach or rent a bike and ride along the beach. Santa Monica is a beautiful and relatively safe neighborhood. I've heard that Venice Beach can be very sketchy at night. If you are around on a Sunday, head down to Venice Beach an hour or two before Sunset and you can take part in this crazy drumming that goes on by hundreds of people prior to sunset.
I don't have a specific hotel in mind for Santa Monica. They do have a hostel, I believe. They also have lots of hotels, so I am sure you could find something.
One last perk about Santa Monica - it is very close to LAX.
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