Facebook lets you take a lot of control over how you want to receive notifications for Groups - as well as your general news feed - whether you want to get a push notification for every update in the group, or receive none at all. You can even choose to only be notified about significant posts, so you'll always be on top of the most important information.
Once you find a group and introduce yourself, don't sit around watching everyone else chat while you become a lurker. Actively engage with the group by contributing links or sharing knowledge. Make sure you comment on other's posts to help get conversations going, which could lead to a more in-depth conversation on Messenger.
If you have job listings that you know about, but aren't interested in, why not pass those along to the others in your group. By sharing the love, you never know who will think of you when they come across a job listing that they aren't suited for, but you are.
It's great to meet and greet with people virtually, but you should consider taking it offline too. Once you've acclimated to a new group and have made solid connections, you should consider organizing a formal evening for everyone to get together. Maybe it's just a small get together with the people in your group, or perhaps you can join forces with other relevant groups in your industry to organize a bigger conference or event. Facebook has built in controls to organize a group event, which makes it easy to plan and figure out how many attendees you'll have, as well as to send out any updates before the date of the event.