Learn French: Cherchons ailleurs, il n'est pas ici. - Let us look elsewhere, he is not here.

Regarder generally means 'to look at' or 'to watch' (directing your eyes toward something). Chercher means 'to look for' or 'to search'. Because the context is about trying to find a missing person, chercher is the correct verb, even though English casually shortens 'let's look for him' to just 'let's look'.

Would it be correct to say cherchons pour since we are looking 'for' him?

No, that is a common trap for English speakers! The verb chercher already has the 'for' built into its meaning. You never use pour after chercher to mean 'look for'. Furthermore, ailleurs is an adverb meaning 'elsewhere', so it simply follows the verb without any preposition.

I've often seen the phrase d'ailleurs. Is it interchangeable with the ailleurs used here?

They are related, but they have very different meanings. Ailleurs (used in this sentence) literally means 'elsewhere' or 'somewhere else' in a physical or abstract sense. D'ailleurs, on the other hand, is a conversational linking word that translates to 'besides', 'moreover', or 'incidentally'.

Why are these two phrases separated by just a comma instead of a linking word like car (because)?