Why is "give me", and not "give to me" correct? As this is dative, and we are told that it is translated (from my language) with the preposition "to" in English.
Here is the scoop on this in contemporary English. There are two possibilities, basically:
1) Give something to someone and :
Give the book to John.
Give it to him.
2) Give someone something
Give John the book.
Give him the book.
So, basically these patterns show that: give is followed by a direct object and then to and the indirect object. Or give is followed by the indirect object, then the direct object.
It is useful to memorize and practice these forms/patterns.
It would be a mistake to think that there will always be exact equivalents or one-to-one correspondences between English grammar and your native language's grammar. It is also not always possible to give reasons: often the learner just has to learn that some verbs follow certain patterns.
As far as give is concerned, it is an example of a ditransitive (or trivalent) verb - it has both a direct object and an indirect object.
UCL Grammar explains:
Some verbs occur with two Objects:
We gave [John] [a present]
Here, the [Noun Phrase] a present undergoes the "action" (a present is what is given). So a present is the Direct Object. We refer to the NP John as the INDIRECT OBJECT.
Indirect Objects usually occur with a Direct Object, and they always come before the Direct Object. The typical pattern is:
Subject -- Verb -- Indirect Object -- Direct Object
Further examples are then provided - in each case the verb is followed first by an indirect object and then a direct object (the same pattern that give typically follows):
Tell me a story
He showed us his war medals
We bought David a birthday cake
Can you lend your colleague a pen?
The grammar explains that
Verbs which take an Indirect Object and a Direct Object are known as DITRANSITIVE verbs
and that sometimes they can double up as monotransitive verbs (e.g. "We bought a birthday cake").
As others have observed, give also allows an alternative pattern ("We gave a present to John").