Here's a website with some iconic locations. I've not been to all of them.
http://la.curbed.com/maps/iconic-buildings-places-los-angelesAlso check the Los Angeles Conservancy, they have a lot of information about historic buildings in LA.
You might want to view the documentary "Los Angeles Plays Itself", it documents the city of Los Angeles through movies and television. It's a little long I thought, but I enjoy seeing LA in film and television.
Here are some places I like:
* Bradbury Building (seen in numerous movies e.g. "Blade Runner"): 3rd and Broadway.
* Disney Hall
* You might want to wander over the the Music Center since you're in the area.
* Los Angeles City Hall, go to the observation deck, free - Mon - Fri 10:00am to 5:000pm. The steps leading into City Hall have been used in many TV shows and movies.
* Grand Central Market
* Union Station
* Getty Museum in West LA, check to see if you need to reserve a parking space.
* Annenberg Space for Photography and Skylight Studios in Century City (some neat modern architecture). The Annenberg is free, and they validate parking, so it's pretty cheap. It's a great place to see photography.
* The Last Bookstore in Downtown LA is an interesting place, but you might poke your head in there, though you can't bring in large bags.
* Our Lady of the Angels Catholic Church was a big architectural project back in the early 2000s. It replaced the Cathedral of Saint Vibiana when that was damaged in the 1994 Northridge earthquake. Saint Vibiana is still there on 2nd and Main Street and it's a restaurant and event space.
* You might want to walk around the Arts District, Little Tokyo, Chinatown, Olvera Street, Broadway (these are all around Downtown LA).
* Clifton's Cafeteria, just reopened.
* For skyline shots, you can shoot over the 110 Freeway at the 3rd Street bridge, there are other bridges over the 110 freeway as well.
* There's some new places that I've not seen before but would probably be interesting, Broad Museum and the Peterson Automotive Museum.
* Griffith Park Observatory is pretty great.
* The Hollyhock House just reopened to the public but they do not allow photography so that's probably a pass.
I have a good friend who lives in Little Tokyo that I visit and I always park in one of the lots, it costs me like $10 bucks for all day (weekend), I just pay it. Depending on where you are there might be free parking on residential streets. Downtown LA is pretty walkable, there are homeless folks around pan handling and they may approach you for change, overall they seem pretty harmless. Also Downtown LA has DASH buses which I believe are free to get around Downtown. I remember back in the 80's and 90's there was a rash of car break in's, I'm not sure how it is now. I feel OK leaving my car in a parking structure, but I wouldn't have anything showing in the car. That's the hard part of going on a long road trip is the amount of stuff you have to leave in your car. If you have a hotel, leave most of your stuff there.