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 ...
My basic advice is: start from what they know.
For example, in some languages, you need to choose different personal pronouns depending on whether they are used as a subject or as an (direct/indirect) object:
He saw me.
I saw him.
"He" (subject) becomes "him" (direct object).
I gave him a present.
"He" (subject) becomes "him" (indirect object).
Declensions add information to the word. This can be gender, object-subject relation and so on.
A language without this feature has to somehow supply this information in another way (think "female steward" vs. "stewardess") (also see Wikipedia)
To learn them the best way will be to listen to native speakers. After a while it will simply sounds wrong if you ...
As a native Czech speaker, I should be able to answer some of your questions, but not all. Usage of the cases is something I have learned naturally (without having any idea of grammar), so I can't help you there. It is also many years ago when I attended an elementary school. Sorry.
On an Android app: You might try Duolingo. I have some experience with that,...
Declension endings in most languages are essential for the use of "correct" grammar. By using tables, you can "calculate" the correct endings. That could be better for "computationally" minded people than just relying on rote memory (which could sometimes go wrong, in my experience).
If you study and memorize tables of declension endings, you will know the ...
In many learning contexts, the aim is for a skill to be so deeply ingrained that you do the right thing without any conscious effort - e.g. a musician hits a key or a string with just the right amount of force, a language learner uses the right tense or ending without thinking, a native speaker writes a word with the correct spelling etc.
But before this is ...