You can try HourHQ from Timekit: http://hourhq.com/ You can use a version of their hosted and branded service for free, but you send your users to their site. Plans that you can host are five dollars per user per month.
Timekit.io looks very cool, but it's $99 a month.
You can find open source stuff for php with MySQL, like http://easyappointments.org/, ...
I am still looking but the most suitable I have found so far are:
Matukio Events from €59
Matukio Events seems to tick all the boxes and looks like the easiest to set up.
JEvents and RSVP Pro @ £62
JEvents and RSVP Pro can track capacity and handle adult/child/etc pricing but looks hard to configure.
Vik Events @ €99
Vik Events seems to tick ...
I have recently done a survey of all the open source resource scheduling and booking applications. See my full list of results.
There are several options that are close or quite close to your requirements. Starting with the most adequate:
Booked Scheduler. My favourite for nearly all use cases, as it's a really flexible piece of software (also making it ...