As others have said, avoid getting the car until you're ready to leave the city. Once you have it, THEN do things like Lombard Street, and preferably on Sunday evening or a weekday when there are less people around.
Twin peaks is great and you'll need a car for that.
Highway 1 is OK, but busy at the weekends. If you want a more interesting drive, take State Route 35 (Skyline Blvd) through the Santa Cruz Mountains. There are several good short hikes on the Peninsular, and several places to stop as you drive down Skyline. California wouldn't be the same without the earthquake experience, and there's one short hike in the Los Trancos open space which has a San Andreas fault trail.
https://www.openspace.org/preserves/los-trancos
I'm not promising anything super exciting, but the trail highlights several geographical features created by fault lines, and in particular the 1906 San Andreas earthquake. It may be worth an hour or so to get out and look around.
Also, I'll second Big Basin. Muir Woods is great, but really busy and can be quite difficult to park. Big Basin is less popular, and still impressive. Again, there are several short hikes you can take from the car which will give you a good impression of the area.
Big Basin Redwoods SP
Continue down 35 to 17, then take that towards Santa Cruz. Again, try not to do that on a weekend or during commute hours.
Monterey and Carmel are lovely. You could easily spend a couple of days there. In Carmel you can head down to the beach and walk quite a distance along it. Coming back along the residential area above the beach there are some stunning homes. In Monterey you'll probably want to spend time around Cannery Row. This doesn't seem to get too busy even at weekends. It's a nice place to stroll around and soak up the atmosphere. The aquarium is a little pricey, but very, very good - worth a visit IMO.
Highway 1 is much more impressive to the south of Monterey as you head down towards Big Sur. Worth the drive, but it's probably still closed south of Big Sur itself. Still, you can drive down and enjoy the sights and a look at the Bixby Creek Bridge.