I did this a few years ago! Or rather, I did half the trip that way. I didn't rent a car but my friends did. The first few days I was on my own, though. Your best bet is to make a list of what you want to see and figure out what's transit-accessible and what's not. You might luck out and everything you want to see has a bus or metro line near it, or maybe you're totally fucked. Really depends on what you want to see.
If I did it again, I would probably consider renting a car; I didn't want to this time because insurance for out-of-country seemed weird, I wasn't super confident in driving, and parking/driving during rush hour seemed like such a huge pain. But you can get around to more parts of L.A. than I expected on just transit. I stayed right by Hollywood/Highland station, so there were quite a few bus lines plus the metro. The bus was surprisingly not terrible, given how much I'd heard people shitting all over them. It does take a while, but it's a bus, what did you expect?
The first few days I was doing stuff like going downtown (Donald Trump had just been elected so I went to the protest) and visiting museums close to LACMA and the California Science Center, and transit was perfectly fine for all that. The worst trip was getting down to the Museum of Jurassic Technology from Hollywood, mostly because I missed my stop and walking between stops in that part of town is pretty lengthy and annoying. I think you can even make it down to Santa Monica via the metro, which is sweet.
That said, there are still lots of places that are going to be shitty to get to. My friends were staying in Silver Lake, and taking a bus from there at night is fine but buses aren't exactly frequent. Getting up to the Griffith Observatory is going to be terrible (though driving is shitty in a different way because of parking). A lot of studio tours are further out too. On one of our last days there my friends and I cruised through Laurel Canyon, and I'm pretty sure you're not doing that in a bus.
I got a week long metro card and got so much use out of it. I was also surprised how little others were using the metro.
Oh yeah, that's the other thing. NOBODY USES TRANSIT. I mean, not literally no one, but it was rarely even close to crowded when I was using it. Coming from packed-like-sardines-all-the-time Toronto, it was blissful.