A large fan of art, I once visited MOCA, the Museum of Contemporary Art in Downtown for an art and photography project. Across the street from the Walt Disney Concert Hall, MOCA on Grand Avenue is both a school for art as well as an art gallery. Similar to the Getty's Villa and her Greco-Roman art, MOCA is the only museum that is dedicated to only exhibiting contemporary art.
I think MOCA is worth mentioning because as a school of art, it not only introduces new forms of art, but exhibits different ways of viewing everyday object. One of the most interesting piece I saw is the structure out front. It appears to be haphazard junk thrown together, but are really intricate pieces made up of different size, shape, and lengths that fit together to make something lasting. Because photos are not allowed, you'll have to go see the place yourself.
MOCA is open to public for FREE every Thursday from 5pm to 8pm, while admission is $12/person, $7 with a student I.D. Parking is a different matter. Downtown is known for it's expensive parking. There are some suggestion on the website such as: parking structures, street parking, and buses. Discounts are available to those with TAP cards.
Not to waste the trip, you should also visit some of the other places nearby: Walt Disney Concert Hall, Ahmanson Theater, Grand Avenue Park