Could you explain the grammar

As for the differences between 가는 거 and 하는 거 we need to break each section down into their individual segments.
    가다 + 은/는 + 것 (the ㅅ 받침 is often removed in colloquial usage) = 가는 거
    하다 + 은/는 + 것 = 하는 거

가다/하다: 가다 is the dictionary form (non conjugated) verb meaning: to go
                   하다 is the dictionary form of the verb generally meaning: to do
은/는: This is a grammatical feature used for indicating. In English, it is most commonly translated as "that".
            ex-  하교에 가는 버스 = The bus that goes to the school \\ 공부하는 학생 = The student that studies.
것 (col. 거): The word for "thing" so you could substitute bus and student in the examples above with the word thing and express this as: 하교에 가는 것 or 공부하는 것 (**It is important to remember that Korean is an extremely contextual language so if you were to only say 공부하는 것 is would be interpreted as "the thing that you study**)

Putting all these together the differences are simply "The thing that goes" and "The thing that does"

Understanding that allows us to move to your second question: 한다 vs 간다
    Notice how in the "verb+은/는" examples the translations are actions that happen in present tense?  Well, with 한다 and 간다 these have an implication of an action that was completed in the past or has already occurred
            ex- 미국에 간 학생 = Students that have gone to the U.S.
                - 공부한 학생 = Students that have studied (while implying they may not study any more)

Now, there is a secondary usage for ~은/는 that is seen in written text such as journals or reports and is simply a way that ends sentences. This is not the same thing as the grammar explained above; however, it leads into your final question of the difference between 저는 먼저 갈게(요) and 저는 먼저 간다

The meaning of both those sentences are the same "I'll go first" but the usage is different.

    나는 먼저 갈게: Your example being lowered speech (반말) it would be more appropriate to use 나 to represent yourself rather than 저. I would say this to my friend if we were leaving a place and I was going to leave first.

    저는 먼저 간다: Is what you would expect to see in written instructions, a journal or a book and with journal writing as well it would make more sense to use 나 in place of 저.

I hope this helped clear things up~