The impressive and architecturally significant Central Library (located next door to the Hilton Checkers) has a fascinating history, and you can hear about it on one of their free one-hour tours. Click the link to see the tour schedule.
"They paved paradise and put up a parking lot" Pershing Square has had many faces over the years. Near the historic Biltmore Hotel (birthplace of the Academy Awards and site of many early ceremonies - walk through and check it out if you can), Pershing Square attracts every type of Angeleno. You may want to park in the lot underneath. In November, Pershing Square will have ice skating! Here is a link for the skating rink: https://holidayicerinkdowntownla.com/ice-skating-downtown-la/ Or you can always just watch. It might be safer. =)
If you have seen Bladerunner, you may be familiar with this building, but it has made appearances in many of your favorite movies and TV shows. It's an architectural icon, yes, but don't miss the Biddy Mason Memorial Park outside the rear/side exit (you can also enter from the street, by going down a few buildings and into the courtyard). It's remarkable.
I haven't been to Clifton's since it closed down as a cafeteria (historic, one of the Green Book spaces in LA), but it seems the Republic is trying to keep the spirit alive. I look forward to checking it out. Open for drinks and bites 6pm-2am Fridays and Saturdays only.