You can connect with individual users by (a) following and (b) messaging them. When you follow a member, you can see updates from them in your news feed (click Following to filter). To follow someone, go to their profile and click the Follow link on the top right of the page. Alternatively, go to the member directory and click the Follow button that appears to the right of their name. You'll see options to Unfollow people you follow and Block people you don't want to follow you in the member directory as well.
You can send a message to another member by visiting their profile. In the top right corner of any profile page, you'll see two options: Public Message and Private Message. A public message @ mentions the member in the public activity stream. A private message can only be seen by the person to whom it is sent.