We've actually been cross-checking advice from here with a guide book and our latest plan is to hire a car upon arrival at LAX and drive to Palm Springs as
PTravel and
azepine00 suggest, staying there for one or both nights. Generic "time near the ocean" is probably being ticked off earlier in our holiday, in either Monterey or Carmel as a side-trip from SF (after driving to Big Sur).
Regarding our first stay, appreciate there's not much in Hollywood itself but the hotel was booked with PC points and we're happy to zoom around on the Metrolink while based there

That's a good basic plan.
And Metro
link is our commuter rail service. Metro is our urban rail system. You want to use the Metro to get around LA... not Metrolink