Your Mail and the Friendslist can be used to a limited extent from the My Iczer Tab, but to really get serious about communicating with your fellow Iczers, you'll have to go to the Mail Tab. Periodically, you'll receive mail here with game updates and news.
Before you can send mail and use your friends list, you'll need friends. There are two ways to add someone as your friend. First, after every battle (including in training) you'll have the option to add your rival as a friend. Second, in the Chat client you'll be able to friend request anyone in the same Chat Room as you. If you have someone you'd like to add, try to meet up with them in the Chat or in a Training Battle. You can arrange the meeting on the Forum.