And the problem is not about the "o-a" distinction only. The same happens to the unstressed "и", for example; personally, I think that the difference is not that obvious as it is for "о-а", but still. E.g. the word "зерно" ("grain") is actually pronounced as "зирно". If we know how to pronounce it but are not sure about the spelling, sometimes we can check the unstressed vowel by recalling other words with the same root morpheme in which the vowel becomes stressed ("з'ёрна" (plural) for "зерн'о", "г'овор" (the noun) for "говор'ите" etc.), and sometimes we just have to memorise the right spelling (for "корова" there's no "cheaty" option). 

What can you do to make it simpler? Well, you have to memorise both spelling and pronunciation, that's for sure. Russian language teachers in schools use dictations; you can try to use different mobile applications - just make sure that there's an audio part there as well so you can compare the spelling and the pronunciation of a word. Keep in mind that an unstressed vowel in the word's root morpheme usually doesn't sound like it's "supposed" to. Actually, it comes with practice, and now that you know about it, it's got to get easier. Good luck! :)

Oh, that's a very common problem, actually. I believe that every Russian language textbook devotes at least one section to the correct spelling of unstressed vowels in words' root morphemes (not sure if it's a correct translation... in Russian it's called "правописание безударной гласной в корне слова").

However, for poor Russian children it's usually the other way around: why do we write "о" when it's clear that we say "а"? The wrong spelling of the word "cow" ("карова" instead of "корова") is a paradigmatic example of children's misspelling. Russian is not "phonetic", and that's a problem for those who (a) don't read in Russian a lot (e.g. little children) or (b) don't speak Russian a lot (e.g. foreign students).

I totally disagree with Vasiliy, it's definitely NOT better "to pronounce like it was written". No, almost nobody says "GO-VO-RI-TE", in reality it's more like "GA-VA-RI-T`E": the third syllable is stressed; the first syllable is shortened. To say it with two o's is as strange as to pronounce the word "speak" as "SPE-AK". Trust me, you don't want to say it this way.

For you, as a beginner, it's much easier to not focus on this. It is better to pronounce like it was written.

This nuance it's like to learn Cockney British accent instead of learning English itself.

In your case, for native speakers both options acceptable.

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When you hear lots of 'o' it does not looks like Georgian. It's called "okan'e" (оканье).  On the other hand  - lots of 'a' instead of 'o' - called "akan'e" (аканье).