Not sure how CA is handling it, in NY when I was given an appointment for the second shot the day I received the first shot. Actually before it was administered. Second shot will be February 25.
Speaking only for myself and as a resident in L.A. County: I got my first shot a week ago today, and when I scheduled the first shot online they also set an appointment for exactly three weeks later for the booster. (I got the Pfizer vaccine.) In L.A. County, at any rate, they have been giving the shots to people 65 and over for at least the last few weeks, maybe a bit more. (That's how I qualified.)
The online appointment system was swamped, and so it was a matter of frequently checking back until there was an opening. The site I used (in Downey, CA) was entirely done as a drive through; I did not even have to get out of my car for it. The process went smoothly and quickly. But as I said, getting the appointment itself was tricky.
A good friend of mine got his first shot at Dodger's Stadium maybe a week before I got mine. Again, getting the appointment required perseverance, but the vaccination process itself was very smooth. At Dodger's Stadium they are administering Moderna with four weeks between shots. I didn't ask him whether he was assigned his second appointment at the same time he got his first one or whether he has to make the second appointment separately.
Edit: I just re-read the email another friend of mine sent me, who also got it at Dodger's stadium. Regarding the booster shot, he said, "We will receive an automatically generated appointment for the second dose via email." That's different from the site I used, which scheduled the second appointment at the same time it scheduled the first.