Learn French: Ce gars travaille avec moi. - This fellow works with me.

Surprisingly, no! In standard spoken French, both the r and the s in gars are silent. It is pronounced simply as ga (rhyming with the 'a' in the English word 'spa').

I originally learned garçon for boy. Can I use garçon instead of gars here?

You could, but the meaning changes slightly. Garçon usually refers specifically to a male child. Gars is an everyday, informal word that translates better to 'guy', 'dude', or 'fellow', and it is very commonly used for adult men.

Why does the sentence use moi instead of me?

Because it follows a preposition. In French, whenever a pronoun comes directly after a preposition like avec (with), pour (for), or de (of/from), you must use a 'stressed pronoun' like moi, toi, or lui. You cannot say avec me.

I always want to translate travaille as 'travels' because they look so similar. Is that ever correct?

No, travailler is a classic 'false friend' for English speakers. It only ever means 'to work'. If you want to say that someone travels, you have to use the completely different verb voyager.