Which one is more grammatically correct: "three pairs of shoes" or "three pair of shoes"?
It depends on who you ask. From the Columbia Guide to Standard American English by K.G. Wilson:
Some commentators object to locutions such as three pair of shoes, arguing that the modifying number should force pair to be pairs. But at least in most Conversational and Informal use, three pair and four pair are appropriate.
Ngrams show that both are in use, although three pairs is heavily favored. To me, three pair of shoes has an old-fashioned ring to it, but I wouldn't correct you if you said it.
As you're describing more than one pair of shoes, you should use the plural form of the noun:
Three pairs of shoes.
Though it's worth noting that if you're describing one pair, this would be correct:
One pair of shoes.
As even though there are multiple shoes involved, there's still just a singular pair.
Some nouns have the same form in both the singular and the plural though. This is something you'll need to learn as you come across the particular words.
Why have one sheep when you can have three sheep?