Visits are defined as being checked in. So, if a customer registers for a class or appointment but doesn't show, this will not count toward their total visits. If they are checked in to either a class or an appointment, they're visit tally will go up.
You can find the total visit count for a customer from their customer profile, just below their name.
For reports that give high level views of customer visits, click Analytics > Reports > scroll to Customers > and checkout these two reports that use visits.