If you are qualified to be a mentor, find a character to become your mentee. Then, press the [+] button under Friend Manager/Mentoring and enter the name of the character that you wish to make your mentee. When the other party consents, you have a mentee.
You can become a mentee yourself if you are level 84 or lower or have not gone through awakening or the 3rd liberation.
If you are not satisfied by your mentor or mentee, you can end the mentor/mentee relationship. Select the appropriate target in the Friend Manager/Mentoring and press the [-] button.