This would be an excellent question for the LA forums, but I'll give you my two cents.. Realize that what we call LA is a very, very big place-- and to a point, lots of places that we may generally call "LA" are in fact totally separate municipalities.. LA County is also quite large incorporating many smaller cities within the county borders..
Southern California, including the greater LA area isn't very user friendly to non-driving visitors.. LA is home to some impressive highways.... Yes, you can get around on public transit- it's not impossible, but compared to other cities of similar size- like Chicago or New York, LA isn't currently as well equipped for mass public transit... What does this mean for you? It means in part that you'd be smart to make some dedicated decision about where you'd like to stay as a "base".. Some like to be in the Hollywood area or downtown.. others prefer one of the beach cities.. there are numerous choices..
So, decide what you think you'll spend more/most of your time seeing/doing, then find a place in that area-- from there you can plan your drives to the other areas that interest you..
There really isn't one place that is convenient to everything.. Just doesn't exist in the Greater LA area.. You can take a look at the LAVCB's site that gives you a good overall feel for the various areas in the greater LA area..