Translation:I have to study!
I have to learn - is also correct.
'I have to learn' = 'Muszę się nauczyć'
It was accepted for me ("I need to learn", anyway)
"I must teach" is it also correct?
No. 'Się' is a reflexive pronoun, so uczyć się literally means to 'teach oneself', i.e. to study.
'I must teach' would be simply 'Muszę uczyć'.