Well, coming from English, look at it like this:

The nominative case is for the subject, and the subject is the one who (or which) performs the action of the verb. The simplest sentences only have a subject and a verb in English: "I breathe".

The accusative case is the case for the direct object, which means the person or thing the verb acts on. "I read the book". "I" is subject, "the book" is being read, so it is technically in accusative case. Like in English, in Russian the accusative noun looks exactly the same as the nominative noun except in two areas: if the noun is feminine and ending in -а or -я, then the accusative is -у or -ю respectively. And if the noun refers to a living person or animal and is masculine or neuter or in plural (any gender), then the ending of the genitive case is used.

The indirect object is less directly connected to the verb. In English, "I give him the book" still has a direct object "the book" which is directly acted on by the verb, and "him" is in what we would call the dative case, which term incidentally comes from the Latin "dare", "to give". We call that the indirect object. Case endings will have to be learned, but you'll find that the -m- sound is actually something even Russian uses in dative case frequently.

You know when to use the genitive in English, as in "father's car". Unlike English, Russian uses the genitive not just for persons (like saying "the door's handle"), and it is used with a lot of prepositions.

The instrumental is used for the instrument with which something is being done, and with certain prepositions, and the prepositive, as the name says, also with certain prepositions and never without.

The endings will have to be learned but it's easier to observe the endings as they are being used by native speakers in order to see their practical usage rather than trying to memorize declension tables and then thinking about which ending to use in which situation.