First a bit of terminology:
declension is a very general term refering to the fact that nouns, pronouns and adjectives are inflected (i.e., take different endings) in order to express number, gender, and case.
declension may also be used to refer to a group of nouns/pronouns/adjectives that follow a particular pattern of inflection (take the same endings to express the same combinations of number, gender and case).
case is the grammatical category that reflects the role a noun/pronoun/adjective plays in a sentence. In many languages, the subject of a sentence is usually expressed by the nominative case, the direct object by the accusative case, and the indirect object by the dative case. In languages such as English, these functions are usually expressed by word order or by prepositions. Moreover, each preposition usually requires that the noun after it comes in a given case.
In general, research about language learning suggests that a combination of conscious concentration on a particular grammatical structure and a not so concentrated exposure to that structure in a large amount of texts and audio materials leads to the best results.
When it comes to cases, conscious strategies might include:
- memorising declension paradigms (such as German "der Mann, des Mannes, dem Mann, den Mann" for masculine singular),
- reading a piece of text and concentrating on identifying which cases are used,
- doing fill-in exercises,
- memorising short sentences or phrases, such as "der Mantel des Mannes" 'the coat of the man'; "Der Lehrer gibt den Mantel dem Mann" 'The teacher gives the coat to the man.'; in this way, you are giving your brain bits of the language in the right context, which makes it much easier for your unconscious to create the right reflexes, and it also makes it much easier to retrieve the information later (the visual image associated with each of the phrases will effectively act as a mnemonic); in my experience, this becomes a highly effective strategy if a spaced repetition software is used for memorising the phrases/sentences.
The less concentrated strategies might include:
- reading a lot of texts, even if it is just beginner level texts in several different textbooks (as long as they contain all the cases that you want to learn),
- listening to slow speech in the language you are learning (e.g., for some languages, radio news for learners are being produced),
- trying to speak (ideally with native speakers) as much as you can, even with errors - quite often, people will repeat what you just said, but with the correct endings, just to make sure that they have understood you correctly.