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What is the difference between these two sentences:

I am going home.

I go home.

I know that people don't usually use the second expression, but can we use it? Also, I don't really know, if it is gramatically correct.

As I understand, in both sentences we are talking about the action which is happening right now. We can ask questions for each of the expressions, for example:

"Where are you going?" "I am going home."

"Where do you go?" "I go home."

So, what is the real difference in the meanings of these expressions?

Good question!
While they seem like they ought to mean similar things, the present continuous and present simple tend to be used in different situations.

The present continuous is used for the specific action that is presently occurring, while the present simple applies more generally.

For instance, if someone asked me "What are you doing?" I would respond with the particular activity I was engaged in at the moment of the question. But if someone asked "What do you do?" I would respond with my job, or the thing that primarily takes up my time.

In the example in your question, "Where are you going?" is the kind of specific question you might ask when you're saying goodbye to someone, or if you run into them on the street. You would expect their destination to be the answer.

"Where do you go?" is unidiomatic, because it comes across as kind of existential without any specific time or activity added to it. However, "Where do you go [to do x]?" is a common formation that you might see if someone were to ask for a recommendation for a restaurant or an activity. Similarly, a question with time added is also common: "Where do you go on Mondays?"

You use the present simple…

…to state facts or general truths. :

Hair grows.

…to express habits or customs:

I usually go home at 6 o'clock.

…to describe a course of action:

First I have lunch. Then I meet my friends. Finally I go home.

You use the present progressive...

...to talk about temporary actions that are ongoing now.:

I'm going home now.

...to talk about a repeated event that takes place in a period including now.

I usually go home by bus, but this week I'm walking home.

...to talk about future plans.

"I'm going home next week."

It wouldn't be fully correct to say "I go home" in response to "Where are you going?". (You would still be understood.) Instead you would say "I am going home" / "I'm going home" or even "Home" (as a some what curt response)

In conversation you would only really say "I go home" when asked generally about something. Such as

"Where do you go after work?" / "What do you do after work?" / "What do you usually do after work?"

"I go home."

Note that this is generally speaking - not relating to the current time. If talking about the current time, i.e. That day, the conversation would be like...

"Where are you going after work?" / "What are you doing after work?" / "What are you doing later today?"

"I'm going home" / "I will be going home" / "I shall be going home" / "I'll be going home"

Note that the questions in this example are asking about someone's immediate plans (for that day). If you wanted to know generally what people did after work then you would use the questions from the first example.

So I suppose as a rough rule, you should only say "I go home" - when you are saying "I usually/generally/normally go home"