Re: What to use to manage club(s) database and classes
To make something "Internet" ready for this purpose requires a "web-hosted" business management solution . While not super expensive, it isn't that cheap either. The ordinary contact management "recommendable" solutions are all silly expensive or useless for this use case.
If you look at something like a professional, web-hosted Yoga Studio or Gym management software solution (seems like a best fit), the pricing is $60 - $100 per month. This includes great stuff like mobile app support for check-in, etc. Still, that is pretty expensive.
Most hosted solutions have free trial versions, you could poke around and maybe there will be a minimally featured "discount" [aka affordable $10-$30 per month] Studio/Gym management solution that could meet your needs with mobile check-in, secure hosted software, adequate reporting, integration with your regular website. The advantages of the hosted solution are security, mobile app support, reporting with server management and backups/disaster-recovery handled by the hosting solution.