Looking at the first part of this sentence without the not only,
Our final goal will be students do acquire the communicative competence
There is something missing from this sentence: you need to use a conjunction that to link the two clauses. You have used should in the second part, and I recommend that you use it instead of do in the first part as well. You don't need a definite article before communicative competence, because competence is normally a non-count noun.
Our final goal will be that students should acquire communicative competence
Adding the second part:
Our final goal will be that students should not only acquire communicative competence, but also learn different things related to society.
Note that not only and but also act as coordinating conjunctions for a parallel structure: They section following them must be the same for both parts. In this case, they appear before the verb in both parts, and students should applies to both of them.
I would also recommend rewording it a little to make it more natural:
Our final goal is that students should not only acquire communication skills, but also learn different things about society.
You might also consider replace different things by more.