You can use you own favourite Calendar program or we highly recommend the use of the free version of TeamUp
A few top tips on sharing calendars and sign ups..
1/ Whatever calendar you use - be sure to link to the 'read only' version that allows your members to view and/or sign up but not 'post' or change items on it.
2/ TeamUp allows you to deploy several calendars - which you could use for different youth activities or indeed volunteer Rotas (all of this is explained in the training videos)
3/ TeamUp also offers the option for folk to 'sign up' for any event or activity (unlike most others we have explored)
This is a link to our 'demo' TeamUp calendar
and this is the link to the training videos on the subject
Below is an example page from TeamUp - but do check out the Youtube explainer to get you started quickly.