If I am Rita's coach, is Rita my *coachee? (yikes) Is that even a word? Would it be correct instead to say she is my ward? What about terms for people at the other end of a mentor, sponsor relationship?
Coachee is indeed a valid word, but often not widely used. I would suggest pupil or trainee instead.
You need to qualify the terms to find suitable counterparts.
If you are Rita's coach (sports) then Rita might be a member of your team.
If you are Rita's guardian (parental) then Rita might be your ward.
If you are Rita's mentor then Rita might be your trainee (in education) or disciple (spiritual).
Update: @Malvolio suggests protégé, which I think is the best answer so far.