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.
Coachee indeed is a valid word.
Alternately, you could use student.
When the word "coach" is used to mean "guide" (as in a business environment), the person being guided is a "follower," but the follower is really an employee. So, as someone who is always willing to push the envelope with language, I propose a new word for the employee who is being coached/guided: emplayee. Now I know this will not catch on, but I just had a little bit of fun. Think about it: when you are an emplOyee, you are going round and round in that "O" doing your job and maybe spinning it a little off-track every now and then. When you are an emplAyee, you are on your way to a lesson from your boss/manager/coach and you want to get an A, like you used to do in school in the classes you loved.