Yes, I've been there a couple of time, though has been a while ago. Nice collection of cars. You can do it in one day.

It is NOT in downtown LA, it is mid-Wilshire district near some other (art) museums. The area is nice enough, not Beverly Hills, but not seedy.
The historic Farmers Market / The Grove shopping mall is nearby and is fun and different.

Not necessary to go to Long Beach for a hotel -- a LONG way to get there. Instead, I'd look due west toward the ocean. Either staying in the Westwood area (near UCLA grounds, also nice to visit) or all the way to the beach in St. Monica area.

Plenty to see & do in the area. What are you interested in?
The Los Angeles Tourism Board has a magazine (online too) with hundreds of ideas, places to stay, etc.