I love saddle clubs. They are family friendly with an overall fun spirit to them. Not to say that they are not competetive - some of them are very competetive. I have found that there seems to be either western based clubs and english based clubs. Just look into which one suits you and your horses. And any club I've shown in you don't have to own the horse.
The saddle clubs here run a series of shows through the summer, with schoolers in the spring and a fun show in the fall. They sometimes host clinics with trainers and education nights with equine nutritionists or vets. Overall they are great! The year end banquet is always a blast too...
There are also special interest clubs, as PP said, like for competetive trail, penning or drill team etc. But to me a saddle club is a group who puts on shows.