I see all of these people with The Patriots, Special Forces, Dead Cell, etc.

How do these groups get started, or how can I join?

Organization Recruitment. You won't be able to join an org yet, because you haven't got the required 100 posts. And you won't be able to make one either, because you don't have the required 1000 posts. Get 100 posts and you can apply for any org.

^Yup, what SSTG said, except you can't apply for The Patriots, they're/we're invite only.

So just to clarify:

100 posts is the minimum requirement to apply for an org (please note that orgs can choose to refuse you, it's entirely the organizations choice on whether to accept you).

1000 posts is the bare minimum to start an organization.

Also you may notice that there are members in organizations that have fewer than 100 posts, this is because they were accepted before the new rules came into place.