For example, I know that StackExchanged has a G+ page, will Genealogy.SE be able to have its own Google Plus or Facebook page/group?
You can create pages and groups, but you'd need to be aware of the Stack Exchange Trademark Guidance.
One of the community managers asked me to remove a logo I had on a Facebook page, since it would violate the terms. Also, you should make it clear that the social media page isn't affiliated with Stack Exchange and belongs to you.
On thing I noticed: It wasn't all that helpful. Someone asked me why I created a Facebook page and tried to drive traffic to it when the goal was to build the Stack Exchange site. It's probably more helpful to find interesting questions that you know others would be interested in, and then tweet or post them using your personal social media account instead.
I generally follow people on Stack Exchange who post their Twitter info in their profile, so that's a good way to network with folks here and be able to share posts here with a much wider audience. Every post has a "share" link below it to help you access the social media tools quickly. We should use them so we don't lose them. :)
Lastly, I know every site does get it's own automated Twitter account, but at the moment, it's just automated. (There was a proposal to grant community control, but I'm not sure the status)
Gene, I think you're more likely to get loads of followers and attention with your personal Twitter account than the SE Twitter accounts will get. Unlike us, they're not real people, and they just occasionally spam anyone who follows them with links. The people who will really be able to help this site grow via social media are the speakers, authors, bloggers, people in the public eye. For instance, when SE's CEO or former CTO -- both avid bloggers -- tweet questions, they get a ton of views.