Check out the observatory during the day, or at night. Both will provide you spectacular views of Los Angeles. Use this as a jump off point for hiking or a better vantage point to see the Hollywood sign.
The new contemporary art museum downtown is a favorite afternoon activity. Follow with a drink afterwards, right next door at Otium. Make sure you reserve tickets online well in advance.
Self explanatory. Park yourself on the beach and grab some rays or catch a beautiful sunset. Or, check out the Venice boardwalk - muscle beach, the skate park, consider a new tattoo. Check out the Hotel Erwin rooftop for a sunset cocktail.
Unfortunately, the rain room is no longer available - but there are always exhibits at the LACMA to inspire. Be sure to check out the exterior exhibits as well (google 'Urban Lights' and 'Levitated Mass').
We have intentionally posted a photo from INSIDE the Getty - to encourage curiosity to explore the OUTSIDE of the Getty Museum (the structure itself is the real draw for us; well that and the always-amazing photography exhibit). A glass of Rosé at the museum cafe mid-museum explore is a must.
There are urban hikes to check out (Runyon canyon, Hollywood sign, Griffith observatory). If you want to find some isolation, but still maintain sight of the city, the San Gabriels will be your best bet. If you want to see the ocean, then the Santa Monica Mountains will be your choice.
Let us know how much time you have, and we are happy to give some recs on the perfect trail for you!