With Compound Verbs
Do not use a comma to separate the paired parts in paired compound subjects or compound verbs.
Incorrect: She lets me watch her mom, and pop fight.
(Compound subject. No need for comma with the word and already there.)
Correct: She lets me watch her mom and pop fight.
Incorrect: They would argue over money, and scream about his late nights.
(Compound verb. No need for comma to separate the words money and scream.)
Correct: They would argue over money and scream about his late nights.
With Subordinate Clauses
Commas do not set off subordinate clauses unless some specific comma rule applies, namely they are clauses in a series, or the clauses are functioning as appositives, nonrestrictive modifiers, or introductory adverb clauses.
Incorrect: He told me that I had better come, so that they would avoid serious trouble.
(Not a series. Not an appositive, nonrestrictive modifier, or introductory adverb clause.)
Correct: He told me that I had better come so that they would avoid serious trouble.
With Nouns and Modifying Adjectives
Do not use commas to separate a noun and its modifying adjectives when the adjectives come before the noun.
Incorrect: The bright red, car was a Corvette.
Correct: The bright red car was a Corvette.
For more on punctuation of modifying adjectives see Nonrestrictive Modifiers, Paired Adjectives, and Adjectives Following Nouns.