"He hates me but I do not hate him."
Translation:הוא שונא אותי אך אני לא שונא אותו.
if you can replace "but" with "rather" you would use אלא in Hebrew. about אך and אבל, they are synonyms, though אך is more formally. btw, there are three more synonyms for but: אוּלָם [u'-lam], really formally and rearly used in everyday speaking, בְּרַם [be'-ram] archaic and probably will not run into it outside the academic world, עִם זֹאת [i'm zot'] is used mainly during the debate about the pros and cons of a particular thing (for example "I don't think I will continue to go out with him, the communication was not something but as we managed to talk, the conversations were reallllyy interesting" that but can be translated to עם זאת)