There are many different ways to livestream.  If you want to use Youtube I suggest you simply google "How to livestream on Youtube" and start learning.  I've never done it.  Our church has it's website hosted by Finalweb and they offer a livestream platform.  We use a program called Wirecast to capture the video and audio (simple camcorder for video and our audio is piped in through our soundboard).  When we stream something it is automatically archived on their website for viewing at a later time.  Additionally, we can record directly to the Macbook running the software.  Tomorrow's livestream was actually recorded earlier this week at several times and will be posted to the site tomorrow morning.