Hey Diane
That should be the easy bit. Really, if someone likes your content and enjoys reading it it should be easy enough to ask them to like you on facebook at the end of each article.
So, we have two obvious social mechanisms we would like to take advantage of:
1. Tweet, Share on Facebook, Google+1
These are the standard sharing icons where we can ask someone to share an article that they have enjoyed via the social channels they participate in. This should help drive exposure and bring more people in as the content comes with extra weight (if relevant) due to the referral from a friend / associate.
2. Follow us on Twitter, Facebook, Google+
People are at a stage where they now understand that they can follow content sources they like and want more of so as long as you have a prominent section on your site which asks people to visit your social page(s) then they can easily click the like button.
Maybe a dedicated part of your page template should have links to your Facebook, Twitter and Google+ pages and you can also use some kind of call to action or hook in your articles like...
_"If you enjoyed this article be sure to head on over to our Facebook Page and Like us over there to get swift access to all of our juicy content" _
Ultimately, you are just reminding people, you can't trick them, they understand how this works now I use Facebook almost as a bookmark and the sites that I enjoy reading, I follow them and when I log into my facebook it is almost a feed of content that I can dip into.
Rand did a great whiteboard friday on improving your calls to action for social channels which is certainly worth a watch:
http://www.seomoz.org/blog/improving-social-subscription-calls-to-action-whiteboard-friday
Also, don't forget the other options, email, rss etc. You can also use a hook to drive people to your email or social channels much like a free (something) that people get for following / liking etc.
As ever, there are a lot of moving parts, play with your articles and calls to action, see what works, push your content notifications out at different times to see what works. Do multiple pushes of content, push older content for newer audiences and just experiment and find out what works for you and your audience.
One last tip, maybe look at the established players in your niche, see what they are doing to drive traffic to their social platforms and work with those conventions.
Hope that helps!
Marcus