When do you use which one?

For a non-native speaking person they probably sound the same. But I guess for natives there is a difference.

And similarly, when do you use

"I should go" and
"I should be going"

I'm a bit confused by this "be", what kind of difference does it express?

You may think that ... -- You may have the opinion that ...
You may be thinking that ... -- It's possible that at this very moment you are beginning to believe that ...
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In general be followed by an -ing word forms the progressive, also called the continuous.
Simple: go, see, do, work, ...

Progressive: be going, be seeing, be doing, be working, ...

The progressive form emphasizes an on-going action.
In the specific case of should go / should be going, the use of the progressive softens the message.
I should go sounds more abrupt than I should be going.
I should go sounds more direct. I should be going sounds more indirect, almost a hint that I want to leave but I want to avoid seeming impolite.

Okay, thanks very much, I got the difference now. But I cannot relate this to

I'll have a drink.

I'll be having a drink.

Because the latter sounds very much direct to me, too, and does not, at least in my understanding, soften. Even more, I feel it's more definite. Is that right?

Note the underlined words.
In the specific case of should go / should be going, the use of the progressive softens the message.
I should go sounds more abrupt than I should be going.
This means that the softening effect occurs with these words, but not necessarily with others.
paul_hI'll have a drink.

I'll be having a drink.
I'll have a drink. -- Serve me an alcoholic drink right now. / I'm going to serve myself an alcoholic drink right now.
I'll be having a drink. -- When you see me (later or tomorrow or ...), I will be in the act of drinking something alcoholic.
The simple tense says what will be done. The progressive tense says what someone will see happening at the moment it is happening.

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Thanks CJ! I think I got it now.

Would you say it's similar with this one:

It started raining
It started to rain.

Progressive - When I came home, it started raining (ongoing action).

Oh, I can't really think of an example with started to rain. Maybe "It started to rain when we were playing soccer"?

While we were in the process of playing soccer (progressive), rain began to fall.

Does that make sense? (Especially the one with it started to rain.)

The contrasts I gave you earlier in this thread are more appropriate for the conjugated forms, as, for example, when to be or a modal (will, may, etc.) is present.
For the case of infinitive (to rain) and gerund (raining), there is barely any difference, especially after the verb start. The gerund (raining) focuses slightly more on the on-going nature of the rain, so the feeling is slightly more, "What a nuisance the rain is!" or "How annoying that the rain keeps going!", but this is neutralized by the presence of start, which happens at a point in time. The act of starting cannot, by its nature, continue.

Regardless, after starting (which already has -ing), use the infinitive.
It is starting to rain is better than It is starting raining.
In short, don't try to imagine a difference between start raining and start to rain. It will just needlessly strain your brain. Emotion: smile