Ok, so there is a scheduling platform out there called Saas. its highly customizable. It allows users who login to your site to request appointments (that you can approve first) and view their past and present schedules and your availablity (if you want them to see that) It can send timed email and even text reminders.
If you're looking at this on a tablet, i havent styled the obile interface, try clicking the "show desktop" link to see what it would look like from a laptop
The free version is clunky and has ads, thus the junk below. For $9 a month though you lose the ads and get almost all the bells and whistles. Most other calendaring platforms start at $30 and suck This is more powerful and is a third the price. I didnt have time to make it pretty, just functionalish, but I can make it seemlessly blend in to any background.