I've seen from the Badges page that we can invite others to join the site. Is there a specific way to do this?
You need to be trust level 2, a "member", in order to do that. From what I've gathered, doing so means that the invitee can respond without even registering, which seems like a nice feature.
Since you easily meet all of the other requirements, you'll just have to show up for the next week or so to get the option.