Learn Italian: Il contadino che lavora nella fattoria è stanco. - The farmer who works on the farm is tired.

In Italian, che is the relative pronoun used to connect a clause back to a specific noun you just mentioned (here, il contadino). It can mean "who", "that", or "which". You would only use chi if there were no specific noun being referred to, because it means "he who" or "anyone who".

Is che acting as the subject or the object in this sentence?

Here, che is the subject. It performs the action of working (lavora). It replaces il contadino for that specific part of the sentence so we don't have to repeat the noun and say "The farmer works on the farm, the farmer is tired."

Why is it nella fattoria when the English translation says "on the farm"?

Prepositions rarely translate one-to-one between languages! In English, we say someone works "on" a farm, but in Italian, you work "in" a farm. Nella is the combined word for in + la (because fattoria is a feminine noun).

What noun does the adjective stanco agree with?

It agrees with the main subject of the sentence, il contadino, which is masculine and singular. The relative clause (che lavora nella fattoria) just interrupts the main idea to add extra information. If you remove it, the core sentence is simply Il contadino è stanco.