What should I learn from the English grammar?

There really isn't much of anything you could call grammar in English, basically just word order, prepositions, and a trace of a genitive case used for possession. If you really want to get any sort of a feel for IndoEuropean grammar you should study Russian or Latin or some such.

If You feel you need grammar then you can go for the grammar. There is nothing like that you should or should not. For those who has not English as mother tongue can have the term should and shouldn't .

To really know English you have to learn:
difference between a and an
plural
difference between adjectiv and adverbs
different pronouns
verbs especially the irregular verbs
when to use do
prepositions
and several other things which you will find in an English grammar book.

You've just described grammar.

There is a lot of grammar in the English language. There are masses of books written about that subject.
To tell someone who tries to learn English that is no grammar is absolutely wrong.

I agree with you. Just as a building requires a structure so does any language. The more complex the topic the more the need for structure which helps one to interpret the meaning of the description. Without structure in language you will not understand complex subjects as the meaning in the sentences would be confusing.