This is a slightly recycled answer, but here we go:

The best way to learn Russian cases is to learn them one at a time, otherwise it will be too overwhelming and a struggle to remember any of them. Tackle one gender, one case, one at a time. Also do singular and plural separately. If too much is covered at once, things will get muddled.

For each case start with some explanation to get a basic understanding of how it is used. Then next comes the important bit. For each ending find some example sentences, which any good textbook should provide, I would try to find at least five, preferably more, and make sure they are memorised. I used flashcards to do this, but choose whatever method suits.

Pretty much all the advice I've read from English speakers who've learned the Russian cases have suggested something similar: drill example sentences.

The good news is that one can use and enjoy learning the language without knowing all the cases. Plenty of mistakes can be made with grammar and the meaning will still be preserved. Even just the accusative case allows a person to construct a lot of sentences correctly.