How do I know which is the object and which is the subject of a sentence?

Basically nouns can be the object or subject of a sentence (as well as other things but we can ignore those for now). The easiest way to decide whether a noun is the subject or the object is to decide what role it plays in the sentence. If it is the thing doing the verb it is the subject. If it is the thing affected by the verb it is the object. For example: In the sentence "Der Hund jagt die Katze" (The dog chases the cat) the dog is the one doing the chasing so it is the subject of the sentence. The cat is the one being chased so is the object of the sentence. The dog is the one doing the verb and the cat is the one affected by it. The easiest way of practicing this is to go over some simple sentences that you might have in an exercise book or on some homework and to try and identify the subject and object in them. Its one of those things that can seem very odd as you never have to think about it in your native language but once you learn it starts to become a lot easier to get your head around.