It means 'I will... if you don't mind" 제가 먼저 말할 거예요. I will speak first. 제가 먼저 말할게요. I will speak first if you don't mind.

It means　"I will~" 말하다 -> 말할게요 I will say 가다 -> 갈게요 I will go Basic pattern is VERB+~ㄹ게요.

Umm i believe it means i'll do for ex. 요리를 할게요.( i shall cook/ i will cook)...however this may not be right so check with a native Korean ^_^