I still don't understand how groups work :(

So, I started up Destiny again, to see if I'd still want to take the plunge with the Taken King. I'm part of the Eurogamer group/clan. But I'm still in the dark how grouping works in this game. I always have been. I've never been in a Fireteam. When I start a mission, do I need to get lucky and see some others in there? Are story missions always solo? I've started the first Dark Below story mission but get killed each and every time. How can I get together with others? I set my fireteam to public always, are there ways to communicate with others when you want to try a mission together? How? I'm lost and alone...