When you purchase a custom domain from WordPress.com, that's exactly what you get - a custom domain. Everything else remains the same, your blog stays on the WordPress.com platform, and you don't gain the functionality of a self-hosted (WordPress.org) site.
At the moment, it is possible to publicize your posts on Facebook, Twitter, Tumblr, Linkedin, and Yahoo!. At this time, Google has not provided an API to automatically post to Google+ and the explanation they've given is that they don't want to see any automated content on Google+. Because of this, we are unable to offer an option to publicize to Google+.
This support article explains the differences between WordPress.com and WordPress.org: https://en.support.wordpress.com/com-vs-org/
If you wish to move, this guide will step you through the process of moving your blog to a WordPress.org instance: http://en.support.wordpress.com/moving-a-blog/#moving-to-wordpress-org