Translation:Normally, the teacher speaks first.
No ,it means that the teacher is the first person who speaks.In one of my grammar books ,in the section of the definite article(le,la,l',les),it says that when someone does something first(premier) or last(dernier) we say "le premier /le dernier".For example "Anne est arrivée la première chez Marc" means that Anna arrived first at Mark's house.That has the meaning "she was the first to arrive",so "la première" there is an adjective,not an adverb.If it was the adverb "firstly" then the equivalent in french would be "premièrement" or "tout d'abord" and it would be at the start of the sentence.The meaning would be completely different though and would need more context to stand as a phrase.If for example you wanted to say "Firstly ,she arrived at Mark's house.",then you would have to say something else that happened next("Afterwards,she....").In that case you would say "Tout d'abord, elle est arrivée chez Marc".("Après,elle...")I hope that my explaining is helpful and not confusing!
"The teacher speaks at first" doesn't mean the same thing as "The teacher speaks first"; it means that the teacher begins by speaking and then presumably goes on to do something else. I think the French is just saying "The teacher speaks first", as in, the teacher is the first one to speak.