A cool new way to discover the city is through the LA art tours. This was basically a walking tour, where the guide took us around the Arts District in the DTLA area. If you like arts and have a penchant for details, then this tour is highly recommended. I took along a friend who was an architect, and she absolutely enjoyed this tour more than I did. The guide knew so much about the artworks, the artists, and the culture of the city. The graffiti art around the city is what interested me the most.
It is a 2-hour walk, so be prepared with good walking shoes and a bottle of water. They also have virtual tours if you would like to do it in the comforts of your home.