Many people advise against specifically concentrating too much on grammar. To be honest, I feel like I have internalized much of the grammar to the point that I don’t have to think about it mainly by reading and listening. What case does “про” take? The one that sounds right.
But! I did study grammar in a methodical manner in school, so I cannot from personal experience say that it is a waste of time – I studied it and I learned it. I think a blended approach works, and that’s what we mostly did in school. Learn about the accusative case, review the endings, then watch for it in your reading and listening. And so on with the other cases. But don’t try to swallow it all at once, and don’t let it get in the way of your vocabulary acquisition. When I came out of school I knew the grammar but had insufficient words with which to employ it.
Maybe I’m a freak, but I found grammar tables useful. That might be in part because my teachers rearranged the usual order of cases to make patterns more apparent. So this might be the only “hack” I can offer: If you do ever want to work from tables, try arranging them in this order:
That’s different than the order in Wiktionary and in most books. The Russian Grammar article on Wikipedia has Prepositional at the end but otherwise in this order. My thoughts on this are mostly around seeing patterns, but the first three here are probably the three most frequently used, particularly in the simple sentences in beginner material.
Some people say that you can be understood fairly well if you don’t get the cases right. This may be true in simple sentences because the usual order of words in Russian and English is pretty much the same. However, because Russian word order is more flexible, recognition of the cases can be crucial to understanding some sentences. My favorite trite example is the difference in these two sentences:
Собака укусила девушку.
Собаку укусила девушка.
Sometimes I have to stare at long, complex sentences in literature or news articles while the grammar helps me sort out the meaning. Adjectives can be separated from their associated nouns by entire clauses, united only by their case, e.g.
Although I say that I “know” the grammar, there are still a few bugbears. Some prepositions can take different cases depending on their usage, and some verbs take a different case than what you might think. It pays to note these differences when you see them.