I have understood so far that Japanese uses many different ways to count things depending on what you’re speaking of. How important are these differences? For example, if I’m referring to two computers, I would want to use “nidai no konpyuutaa” 二台のコンピューター, would it be grammatically acceptable to instead just use “futatsu no konpyuutaa” 二つのコンピューター?
These counting systems are scaring me!
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There’s no “one size fits all” answer for this. For a large number of things, つ will be fine. For other things, it will sound unnatural… that is… no native Japanese person would resort to 二つ for certain things. For instance, animals would never be counted with つ. And if something has a super obvious and basic shape that has a counter, like “long and thin” or “flat”, then つ will be less natural.
But at the end of the day, you would never be misunderstood. And using vocabulary is always a continuous path toward “more natural.” It’s almost impossible for a non-native to ever reach “100% natural” vocabulary usage 100% of the time, and so fretting about being unnatural now and then isn’t a good use of time or energy.
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Ultimately I try to understand the ～台 not because I want to use them correctly I want to understand them when said. So I will be learning all that I can. The good news is there is a ton of repetition in the numbers used so for example since I can already count to ~9,999. I can also count all machines and understand the counting of all machines to ~9,999. Now just if I can count people, days, and things to ten… (those are definitely being leaches to me).
I think they will always give yes credit for TRYING!
There are so MANY counters and the number pronunciation changes (sometimes unpredictably, it seems to me). We can only try.
10,000 is 一万. And in the same way as we number thousands in English - i.e. 310,684 = (three hundred and ten) thousand (six hundred and eighty four) - in Japanese, you cound the number of ten-thousands by putting the number before 万, and the leftovers after. So, for example, 12,345,678 in spoken Japanese is grouped as 1234,5678, or 千二百三十四万五千六百七十八 - i.e. (千二百三十四 )万 (五千六百七十八 ). (Though, it’s still written as 12,345,678 if English numerals are being used.)
And now you can count to 99,999,999. 
The next power of 10,000 is 一億, and in the same way as we count our millions separate from our thousands, Japanese counts 億 separate from 万.
123,456,789,012 = (a hundred and twenty three) billion (four hundred and fifty six) million (seven hundred and eighty nine) thousand (and twelve)
= 1234,5678,9012 = (千二百三十四 )億 (五千六百七十八 )万 (九千十二 ).
And now you can count to 999,999,999,999. 
That’s as far as you’re ever likely to need to count, but if you’re curious, further powers of 10,000 go 一兆, 一京, 一垓, 一𥝱, but I doubt you’ll ever need to know those - think I’ve seen 一兆 maybe twice, and my IME doesn’t even recognise the higher ones. But I digress…
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There are also natural groupings that help as well. For example:
「千」「二百」「三十」「四」for 1,234. Since the breakpoint is 10k, knowing how to count to 9,999 gets you 99.99% of the way there. 
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There was just recently another pretty good thread on this topic!
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Not actually related to counting, but near where I live there is a station called 十三駅.
It’s not じゅうさんえき, but rather じゅうそうえき.
I think they just want to keep everyone on their toes.
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Ah, you of all people know that place names break all the rules. 
Here’s some gems I discovered:
五十島 = いがしま
五十川 = いらがわ
五十市 = いそいち
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