Whether it is correct to use a comma before a coordinating conjunction (“and”, “but”, “or”, “nor”, “for”, “yet”, “so”) depends on the situation. There are three primary uses of conjunctions:

When a coordinating conjunction is used to connect two independent clauses, a comma is always used. Examples:
I hit my brother with a stick, and he cried.
The rain stopped, and the sun came out again.
Should I eat dinner, or should I play a game?

When a coordinating conjunction is used to connect a dependent clause, a comma is never used. This includes both of your given examples. Other examples:
The boy ran to his room and cried.
Frank is a healthy and active child.
Should I eat dinner or play a game?

When a coordinating conjunction is used to connect three or more items or clauses, a comma is optional (though I personally prefer to use one). Examples:
I bought cheese, crackers, and drinks at the store.
Should I eat dinner, play a game, or go to the store?