i just got back from a vacation there myself, so i'll throw in my 2 cents in respect to an outsider...
Rent a car without a doubt...it'll cost a bit, but it will pay off by having access to all areas of the city and surrounding areas.
As far as where to stay, we stayed downtown...which I know has kind of a bad rap, but it was definitely pretty good for its purpose...we didn't intend on spending much time where we stayed and didn't...we stayed at a hostel for $34 a night...3 people/2 twin beds...ended up being around $12 a night per person...plus $7 parking a day as well. For its purpose, it got the job done...if you're looking for a nicer place (an actual hotel), you may try the hollywood area i suppose, since there's so much tourist stuff to do around there.
Like I said though, staying downtown was a pretty good location because it was located within about 20-25 minutes of anything we wanted to do...we did Hollywood, Santa Monica, Venice Beach, Beverly Hills, UCLA, South Central, Staples Center. We also went to Manhattan Beach and Hermosa Beach, which is a little bit further of a drive, but still not too far.
As far as timing, I'd say your best bet is to stay out later (1:00-2:00 am) assuming you're with friends, and sleeping in til like 8:30 am or so, then getting ready for the day...that way, by the time you're ready to set out for the day, it will be around 10:00 am and the traffic will have settled a bit.