What is the difference between 나 and 난 and 내가?
Feel free to just provide example sentences.

나-me, I

난 is actually from 나는

나는 and 내가's difference is quite hard to explain

when we use ~가/이 we explain what subject does
so when I put myself before ~가, I see myself in the third person's point of view. 가/이 are used for present state or something which is happening now.

내가 할게=I'll do it(right after saying, I'll do)
선생님이 가신다=Teacher goes somewhere(teacher is going somewhere right now)
엄마가 말씀하신다=my mom says(mom is saying sth)

on the other hand, 나는 is used to explain state or characteristic of a person. so 은/는 is used for something doesn't change normally.

나는 간다=I go(I go but I don't know when)
남자는 말해=a men says(a men says but not now)
선생님은 친절하시지=my teacher is nice(teacher is nice always. not just now)

can you get it? you can use 은/는/이/가 when you make sentences but 이/가 is more used for spoken sentences