Why don't we use commas after conjunctions in some cases, when we usually put commas between two independent clauses?
We went to Bar Harbor but did not take the ferry to Nova Scotia. (Why there is no commas before "but"?)
I often go to the seashore and collect rocks there. (Why there is no commas before "and"?)
You do use a comma for two independent clauses, but you do not use a comma for a compound verb.
An independent clause is a sentence-like clause that has complete meaning on its own and can stand on its own. The part of the sentence in bold is an independent clause.
We went to Bar Harbor but did not take the ferry to Nova Scotia.
Consider the second part:
did not take the ferry to Nova Scotia.
That is not a complete sentence. It does not have a subject (“we”). Therefore no comma is needed, because what we have is a simple sentence with a compound verb, not a compound sentence with two independent clauses.
The case is the same with your second example. The compound verb is “go and collect.” There are not two independent clauses in that sentence either.
If you had two independent clauses, it might be something like this.
I often go to the seashore, and Jack collects rocks there.
In this case, you have two separate clauses. They are combined by the conjunction “and” and a comma. When you have two independent clauses, you must have two distinct subject-verb pairs.