I agree with Ramil that there is no easy way to learn cases -- Russian grammar is very complex and will take time, but don't let that discourage you! Here are my tips as a non-native learner:
1. Invest in an actual textbook/workbook, or take a class. Actually writing out exercises will help you remember much better than simply looking at a chart.
2. Be familiar with the terminology of your own native grammar first. Know how to identify things like the subject of a sentence, a direct versus an indirect object, etc. With grammatical cases, the words change depending on what role it is playing in the sentence (essentially, the difference between "I" and "me," but with everything and 6 different ways!), so coming in with that basic knowledge is very helpful.
3. Work on fully understanding one case at a time, with all of its endings, spelling rules, and common exceptions.
4. Work on your Russian a little bit every day, even if just for a short amount of time.

Let me know if you have any specific questions about cases or the process of studying Russian, and good luck!