You were right in a way. It's the conjunction and that's a connector. The comma is a separator.
The general rule of thumb is to add a comma only if and joins two sentences (a sentence has two things: a subject and a predicate. A predicate is everything that comes after the subject). If one of the sentences is missing its subject, then don't use a comma.
1. Alice read the book, and she saw the movie. comma
2. Alice read the book and saw the movie. no commaAlice and she are the subjects in 1. That sentence has two subjects, which means it has two clauses, so a comma is generally added before and, not after it. In example 2., there is only one subject, Alice, so a comma is not added.