With radicale, do I need to install some other somewhere in order to use it?
No, you just need to install Radicale. That's it. calDAV and cardDAV are widely used formats available as an option with basically any calendar.
Can a self hosted calendar still send and receive invites to other calendars?
Oh, I see your problem. You don't host your calendar. You host a service that is used to synchronise all the regular calendars you already use over different devices.
Or are you at the moment using Google's calendar in browser only?