Your Friends List can be found on the right side of the Core Client/Games Browser:
To send a friend request to another user, click on the "Add Friend" icon near the upper-right corner of the screen, at the top of your Friends List:
After clicking the icon, you can send a friend request by either:
a) Searching for a friend by username, and then clicking the "+" (plus sign) next to their username, or
b) Click the "+" (plus sign) next to a username
The displayed usernames are those that are "Recommended" or for those you have "Recently Played With" (including the name of the game you played together).
At the bottom of this list, you can view your sent/pending invites that have not yet been viewed/accepted.
After your friend request has been accepted, your friend's username and status will display in your Friends List.
Once a friend has been added to your Friends List, simply click on their name to send them a message: