The vast majority golf groups are put together around social clubs, neighborhoods, or other common interests. Start networking in your area, asking friends, neighbors etc if they play in or know of any golf groups. Also chat with the folks at your two starter shacks. (Networking)
We found playing with groups that are generally close to our area makes it easier when trying to choose courses and where to eat afterwards. Many groups are as socially oriented as they are golf oriented. Don’t worry about your level of play as long a you play “ready golf”.
TV does have organized weekly golf events for both men and women. Those can be more serious and usual have prizes/awards.
I’m sure other will chime in.
We each play a couple of times week individually and in two couple’s groups.
Don