Are you learning Russian to "get by" on a one-week business trip to Moscow? Or do you want to learn "good Russian"?
Russians will love you if you speak "good Russian." Educated Russians are extremely proud of their grammar skills, and rightly so. Russian grammar is unbelievably complex. The Russian word for "good" sounds like "horror show," and if you want to learn "horror show Russki" (good Russian) prepare for some scary stuff!
If you learn Russian grammar, you'll see how precise a language can be, and how imprecise English is in comparison. English is full of useless little words such as "a," "an," "the," "this," "that," etc. These words add no meaning to sentences, they just make sentences "sound right." English also conveys meaning via word order and context. Russian words communicate meaning so clearly that the order of the words can be mixed up. For example, in Russian you can say "Bob eats lunch" or "Lunch eats Bob" and there's no doubt that Bob is doing the eating.
Pronouns ("I," "you," "they," etc.) are built into verbs, so Russians often drop pronouns without losing clarity. Thus Russian sentences are typically shorter and less ambiguous than English. The price you pay for this efficient, precise language is far more demanding grammar.
But to "get by" you need only basic grammar, but not the byzantine grammar of "good Russian." You could treat all nouns as if they were masculine, and all verbs as if you are the person doing the action, and Russians would usually understand your meaning. But you should read over the many grammar rules so that you have a clue what Russians are saying. E.g., you should be able to recognize when a Russian uses the prepositional case, even if you only use the nominative case.
Native speakers learn grammar as children by listening to adults talk, and by being corrected by their parents. A child who reads a lot and whose parents speak correctly doesn't need to learn grammar rules. As an adult learning Russian, you'll learn best if a native Russian listens to you and corrects your mistakes. But the grammar rules will act as shortcuts, to help you learn faster.
In learning some people are auditory learners, some are visual, and some are movement learners. (See ISBN 1573240648 "The Open Mind" by Dawna Markova, for more about this.) But all three learning styles are needed for organizing and committing to long-term memory. You may prefer to hear spoken Russian, or see written Russian, or (for movement learners) write a Russian word and then write how it sounds in English. You may need to do an activity, such as cooking dinner, to pay attention. But all of us need to do all of these things to learn well.