The sentence that shows a relationship between two independent clauses is: "When Brian gets bored, he goes to the internet, and he looks for a game to play."
However, the original sentence has some issues with punctuation; it can be improved for clarity. The independent clauses are "he goes to the internet" and "he looks for a game to play."
If you are looking for two independent clauses joined correctly by a coordinating conjunction, you might consider rephrasing or correcting the sentences.