Learn French: Les cochons sont dans le jardin.

Both are actually correct! However, dans le jardin strongly emphasizes being physically inside the boundaries of the garden. When talking about animals like pigs getting into a specific enclosed space, dans le is the most natural choice.

Do we pronounce the 's' at the end of "cochons"?

No, the s is completely silent. In spoken French, you usually tell that a word is plural by listening to the article before it (like les instead of le). You would only hear a plural s if it linked to a following vowel, but here the next word sont starts with a consonant.

I sometimes mix up "sont" and "ont" when I hear them. How can I tell them apart here?

Sont (are) always starts with a sharp "s" sound. If the sentence used ont (have), the silent "s" at the end of cochons would link to the vowel in ont to make a buzzing "z" sound (a liaison). Because you hear a sharp "s" sound instead of a "z", you know the verb is sont.