Thanks for asking about book clubs.
Although I've visited around 100 book clubs over the years, I'm not much of an expert in how to form one. What I can say is that most book clubs meet once a month. Some are very informal, where the books are mainly a reason for a social event (but what a good reason!).
Some are more formal, with organized discussion of a book. The most common approach is that each month one of the members will pick a book to be read and discussed the following month. Sometimes that person will pick a few titles and people vote on their preference. Each month, the member who chooses the book rotates.
Then everyone reads the chosen book and prepares to talk about it at the next meeting. Once in awhile - if the book is by an author who isn't too far away - the author of the book is invited.
Of course, there is always food and usually wine. Sometimes, it's potluck at a local clubhouse. Some clubs meet at restaurants. Other clubs will rotate houses and the host provides food.
The average book club is about 8-12 people. A few are more than that, although more people makes things less organized, louder, and less suited to discussion. I've been to some clubs with only 5 or 6 members. Once in awhile several book clubs will get together at the same time to host an author at a larger venue.
Organizing a book club is as simple as inviting people to join. Often but not always, there will be some kind of general focus to the club, like mysteries or romances or literary or "anything goes." I've been to one club that chose from all genres except romances. (I suppose that was because romances are the most popular genre and the club wanted to expose readers to other genres.)
One of the obvious attractions is that in book clubs people will meet other people who are highly intelligent, educated, and interested in an intellectual world. I've been to clubs where people became best friends with each other. I've been to clubs that have been meeting continuously for over twenty years.
Another attraction of book clubs is that they generally don't have hierarchy and officers and minutes and raffles and pledges and all the other stuff that service clubs have. Book clubs don't "come to order" and have motions etc. They are simply fun gatherings focused on books. (Please know that I very much appreciate what service clubs do. I'm just pointing out differences.)
I hope this helps.