https://wonderfulwanderings.com/week-in-los-angeles/
https://www.smartdestinations.com/blog/los-angeles-vacation/
What are your interests? What are the things you’ve enjoyed visiting in places you’ve been on vacation?
For me a day at the beach is fun. Even a couple of days! If you’re a beach person you could go to Hermosa or Manhattan beach for a day, rent a bike to ride the bikepath for a while, relax on the sand, walk around the smallish downtowns a bit. Or you could visit Santa Monica and go to the 3rd Street Promenade, down to the beach to relax on the sand, or rent a bike to cruise around down to Venice. The weather in October in LA is likely warmer than you expect: https://weatherspark.com/m/1727/10/Average-Weather-in-October-in-Santa-Monica-California-United-States
A day trip to Santa Barbara is another idea.
Outdoor patio dining and drinking at Nobu in Malibu. 
Tar pits.
No other world city – not NY, London, Paris, or Tokyo – has tar pits that are even remotely comparable. LA’s tar pits are in a class by themselves. Just be sure to watch your step.
Griffith Observatory is definitely worth a visit. And I agree with both the Getty Center and the Getty Villa in Malibu. The Getty Center in particular has wonderful views and outdoor spaces in addition to the art. I highly recommend taking the architecture tour there.
I second the Griffith Observatory and suggest going in the evening. The view of the city lights is spectacular.
I think you may enjoy staying in Santa Monica at the end of your trip, although I don’t think they allow rentals. But there are some nice hotels/motels if you don’t want the expensive ones right on the beach. Santa Monuca is so nice and more laid back. You can walk the beach, disntiwn Santa Monica is adorable. Fairly close to Malibu. I’d stay there.
Thanks for all the information…bumping to top for further feedback! I’ll definitely be renting a car as the area is so large—so many things to see, so many good suggestions!
I’d appreciate suggestions on favorite accommodations…
Griffith Observatory at sunset was great. Hard to park, but still great
Getty Center - take the architecture tour. Well worth it and more intereing to me than the art.
You can rent bikes in Hermosa Beach (south of LA) and ride south to Redondo beach along the “strand”. Beachers are wide and less crowded than the
Broad museum and LACMA if you like art.
Walking along the Venice beach boardwalk is interesting.
If you have any interest, there is also Disneyland.
With an entire week, I agree with some of the day trips (or overnight) - San Diego, Santa Barbara etc.
I’ve attempted to go to the Griffith Park Observatory 5 times in the last year or so, and each time the closest parking was over a mile away with the walk going uphill, and/or cars were being turned away. Weekday, weekend, didnt matter. So I wouldn’t count on Griffith Park much. There might be a shuttle at the bottom parking lot that might be available but I never tried it before.
Plan on everything taking a lot longer than you think it will (driving, parking). It can be “hard” to do a lot in LA! I second the recommendation for Manhattan Beach or Hermosa…we find these a million times better than Santa Monica. We’ve tried SM a couple of times and the traffic, parking and crowds are just horrendous, plus the beach at times is dirty, smoggy, and not nice. If you want to drive up a bit, Malibu and Zuma are gorgeous beaches. Manhattan is closer to the airport if you are flying into LAX.
Where to stay really depends on what you think you might do…LA is so spread out that nothing is “central”. There are loads of Airbnb options, many quite interesting.
I just came back from a trip to CA, including Santa Monica/Venice and it’s much more crowded than in the past. Hop On and other tour busses, the rental bikes, and tons of people. (Granted, this was a Saturday.) They also changed to make street parking more difficult, imo. So still consider avoiding the car and using Uber/Lyft.
And several friends said the drive from SM into Orange County is a wall of traffic. That’s just half the trip to San Diego. I just looked at the google map and it doesn’t look that bad (830 am in CA, 2.5 hours, all the way.) But you may need to time this just right. Check the maps in advance (ie, before you go) for traffic on different days, at various times.
sunset blvd west to Temescal Canyon…turn left and go down towards the beach…It is beautiful. Robinson Gardens in Beverly Hills (you must make reservations. Venice beach. Disney Hall. Sunset Plaza…very chic…great window shopping.
Universal Studios, Disneyland for sure. There are other stuff to do like explore downtown Los Angeles. The LA county fair just ended last week sadly. Orange County is not too bad traffic coming in on weekends. Use local arterials if you have to to get to OC. Santa Monica to Irvine is about an hour on the weekends. During the week you should give yourself an extra half an hour, which is not that bad. If you want to get to San Diego using the toll road State Route 73 between 405 and 5 may save you some time on the commute. Rates are about $5-10 one way and you pay within five days online when you make an account.
That trip I just took was a drive down Rt 1 from north of SF, ending at LAX. Trust me, you want to see the ocean and as much of it as you can. It’s mindbogglingly beautiful. I actually went on blue panorama overload (and after living there previously, for so many years.) I truly think sticking to LA county for other activities is fine. That incudes Santa Monica and the Venice boardwalk, Griffith Observatory and the rest. If you’re up for it, treat yourself to a fancy meal or two. (That may need a @bevhills recommendation.)
If you’re into music the Grammy Museum was fun, well done, and not crowded at all.
Add me to the list of fans of the Norton Simon Museum in Pasadena. It might be my favorite small city art museum. They have a huge collection of impressionist paintings.
Here’s some info on ways to get to Griffith Observatory, including an inexpensive shuttle bus that runs from the Vermont/Sunset metro stop on the Red line every day of the week (note that the Observatory is closed on Mondays):
http://griffithobservatory.org/visiting/directions.html
I haven’t been there for at least a few years, but it’s always difficult to find parking close by. Be prepared to hike uphill some distance if you do drive there. It really is worth a visit though.
I am also a huge fan of the Norton Simon Museum, and nearby Old Town Pasadena is a fun place to walk around. There are lots of shops and good restaurants there. And if you do venture to Pasadena and have any interest in architecture, I highly recommend a tour of the Gamble House. It was built by Greene and Greene in 1908 and is a wonderful example of American Arts and Crafts architecture.