Are any of you cyclocross riders? How do/did you get started? Are there any weekly recurring rides near West LA/Santa Monica?

Well, the season just ended.
Starting in late August or September; https://www.socalcross.com will fire up their clinics with races starting in mid to late September most likely.
I'm pretty sure there was a westside training group this year.
Also check the socalcross facebook page, it gets more updates.

I got started by going to one of those clinics.
You can use a MTB to race if you want - or go all in and get a CX bike.

Cool, thanks for the info. Is there a single speed category for CX? I was thinking of building a single speed CX bike that could double as a commuter...

This year there were Single Speed A and B categories for the men. A single division for the women.
You can also race a SS in the other classes, just suffer a bit more.

I just bought my first bike in a long time and I ended up buying a spec. I am starting to venture into the mountainous areas for some strength training. I wouldn't think a single speed for cx is all too wise since it's such a versatile bike that you wouldn't want to limit your possibilities, but hey, that's just me.