My father trained me not as a man or a woman, but as a performer.

Literal

Father [topic-は] me [object-を] man or woman [as-として] [not-ではなく], performer [as-として] trained.

~としてではなく、~として — 'not as X, but as Y.' ~として (in the capacity of) appears twice for contrast. 演技者 (performer) is a formal term.