I have a question about this character: 雨
I know each kanji has so many pronunciations, this is pronounced "ame" but it has other pronunciations, could you please tell me when to pronounce it "ame", when to pronounce it "ama" and when "u".
At its core, the difference is that う is the 音読み, and thus is used in 漢語【かんご】 (words of Chinese origin). あめ and its related forms are 訓読み, and are used in combination with other readings of Japanese origin.
Now to get into some specifics...
あま is related to あめ, and is used when あめ appears at the front of a compound and a vowel shift is required. For example:
さめ as in 小雨【こさめ】 is another sound shift-based derivative, this time for when it appears at the end of a compound.
あま is also a reading for 天 (which has to do with the sky in this case), however this is in fact distinct from 雨 in nearly all cases. A review of matched pairings in 大辞林 produced just three such cases, each of which is close-but-not-quite:
う is the Chinese-derived reading, and as such is used in 漢語. The 大辞林 entry on 雨【う】 provides several meanings and example uses:
Examples of happiness or blessings
Examples of things that fall like rain
In short, poetic usages seem to consistently fall under う. Beyond that あめ or one of its derivatives is your safest starting point when guessing on an unknown compound.