How is it going?

To that question, should you answer I'm going great or it's going great?

Is it right to say something is "going" great in the first place? I feel like I'm using it to mean I'm going somewhere....

Just answer "hello"

Everything is fine.

No... don't say "I'm going great". That is just french. LOL

"It's going well" is what I usually say.
Actually just "good" will do.

Most native speakers simply say "good" or "not too bad".

So you could just say "great".

Say "how's WHAT going?".

In "It's going great", 'it' refers to the present situation or life in general, and it's a common response to greetings like "How's it going?". "I'm going great" doesn't make much sense.

"I'm DOING well" (or colloquially and more emphatically, "I'm doing great") is a more likely response to "How are you doing?"

So, you do have to pay a little attention to how the greeting is worded to figure out how your reply should be worded -- they should agree as much as possible.

"Going" in this sense is a common English expression that can mean not so much the literal sense of "moving in a direction" but also the more abstract ideas of events or situations progressing, happening, and unfolding. So don't be weirded out by the literal meaning; it's not what's being used in this case. It's also not limited to greetings; similar questions might be, "How did your presentation go?" "How are your studies going?" or "How's it going with your new job?"

Uriel, thanks for your post.
To "How are your studies going?", I once replied "They are going great". The actual answer has to be "It's going great" even if the question asked about plural nouns, right? I'm so embarrassed now that I know what I said was wrong.

No, "they are going great" is perfectly fine, as the noun is plural. There's no reason why plural nouns can't be "going great".