My time in LA was for my thesis, scouting places where the cinephile character would go simply because of their use in his favorite films, so it was pretty touristy in its own way (though not many people flock to Koreatown to see/eat at a restaurant used in Chinatown).
I stayed in a cheap and clean hostel in West Hollywood, and fell asleep to bucket drummers and a ripoff business that sold the idea of driving a Lambo and Ferrari to tourists by revving the engines, but some places/things that were cool:
- People-watching in West Hollywood. Anywhere really, but there's a reason they're called Hollyweirdos.
- Public transpo. Slightly limited by comparison to the size of the place, but very useful and cheap, and sometimes something odd and incredible happens.
- Farmer's Market at The Grove
- Look for "peephole cinema" installations. I found one in an alleyway behind a restaurant in Chinatown. It was playing an experimental film on loop.
- Getty/LACMA/Museums in general. Also the Griffith Observatory. Barnsdall Art Park
- Runyon Canyon. The kind of spot where lots of people go for a mild hike. Occasionally somebody famous is spotted during their morning run, but some cool sights at the top.
- See a movie at New Beverly Cinema, at at the Hollywood Forever Cemetery
Some people don't advise walking around, but I enjoyed doing it. I walked from West Hollywood to Little Armenia to Koreatown, and just saw so much interesting stuff.