When I played, you signed up at your local course and posted your scores. Back then I think it took 5 scores to establish your handicap.
It hasn't changed. It's been modified a bit to fit the times, so to speak. @Bengal B building on what he said ... If the place has a club pro it's going to be a USGA member club. You're going to pay dues and pay for their handicapping services the vast majority of the time. He's right, five rounds. They require complete scorecards and you'll enter each round. You're first score is calculated at five rounds, then every 20 thereafter. It's goes to the point of being able to get an official handicap card from the USGA. There are qualifications for that as well (events participated in, etc.) and of course, more money due. tl:dr Years ago the same management company operated three courses here; Crowfield, Patriots Point, and Pine Forrest. That was a deal I went through that process at Crowfield and stopped keeping track a few years later. I carried a nine.
I've played hundreds of rounds and not until recently have I ever turned in a card. I think I shot a 94 I figured I may as well start the process and beings I am now a member of a club I should do what all the other members do so that when I enter a tourney there isn't much bitching going on. The greens are FAST, the sign says 12 but I bet they are rolling at least 14, its a bunch of kids doing it and I doubt that they actually check. If you screw up and wind up on the high side you had better drain it coming down otherwise you are looking at a chip back the other way. Played 2 days ago and had 16 putts on the front and 14 on the back. That was a good day. Brutal
