Learn English – Is this “not only…but…as well” sentence correct

The first suggested alternative is not grammatically correct. The possible orderings of "you can not only" are:

• "you can not only"
• "not only can you"

Also, there are not strict limitations on how much of the second clause you may/must repeat. The following are all grammatical:

"you can not only verb object A, but also you can verb object B"
"you can not only verb object A, but also verb object B"
"you can not only verb object A, but also object B"

All these sentences would be correct with either "not only can you" or "you can not only".

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.