Hiragana are used for native Japanese words, but also very commonly for kanji, even for onyomi.
Readings of kanji can be split up into 3 broad types:
On'yomi
These are readings taken from the Chinese mainland when kanji were imported to Japan. The reason why there are so many is that kanji sometimes took readings of chinese characters from different regions of China (where different dialects exist), and at different time periods. 走, pronounced "zou" in Chinese, was loaned to Japanese as そう, and also as しゅ. There aren't any words with the しゅ reading that I can actually think of. It exists, but it's very uncommon, if it's even still used.
I have more info on seemingly "irregular" readings of on'yomi here [Apparently unrelated 音読み? ] if you want some more reading about this.
Kun'yomi
These are Japanese readings of kanji. They were meant to match the meanings of the chinese characters when they were loaned to Japan. For instance, 走 in classical chinese means to walk or run. The japanese word for "to run" was attached to this character when people started writing with kanji in Japan, so 走{はし}る, meaning "to run" is written with this character.
The reason why there are sometimes many kun'yomi is that there are many Japanese words which are similar in meaning, so take the same kanji, but different in pronunciations. 降る is one example. It can mean "to fall", or to alight / get-off.
雨{あめ}が降る{ふる} (It's) raining [literally "rain is falling"]
バスを降{お}りる Get off a bus
Because of this, lots of kanji have many kun'yomi too.
Other readings
Some kanji are given new readings when in words together. 下手{へた} is one example of this, or even 上手{うま}い meaning "unskilled / bad" and "good / skilled" respectively. Neither of the kanji normally take either of the pronunciations given here, but when you put the two characters together, it makes this reading.
Another example is たばこ (a loanword "tobacco"). Even though it's a loanword, it's often written in hiragana, or even Kanji as 煙草{たばこ}. Here, the loanword was given kanji that match its meaning, even though again, neither of them naturally are pronounced like any part of the word "たばこ".
More recently too, you'll see that Chinese names of places will be accompanied by neither on'yomi nor kun'yomi readings. 馬英九, current leader of Taiwan, will have his name written in furigana above these kanji as マー・インチウ, even though they could (and still can) also be read as ば えいきゅう, and the name is even Chinese, from which kanji were taken.
Other side points
You were talking about other alphabets, and while you're not wrong, that's not the whole story. Katakana is used mostly for loanwords, though not even all of them, as we saw with tobacco, and sometimes hiragana will be used stylistically by some people or companies. While kanji readings can be written in hiragana, I've seen some people use katakana for words like 綺麗{きれい}. It's also used for names of animals and plants in science, such as イチゴ (strawberry). But, kanji are also used for names of living things too, and in some cases, so are hiragana. Katakana is used scientifically though.
Knowing which to use is usually down to context, but really needs to be done on a case-by-case basis. Not even Japanese people know all the time how to read somebody's name without being told (and the opposite way round -- knowing how to say it, but not how to write it without being told).
Still, I don't want to discourage you from Kanji. They are my favorite part about Japanese, and they're more helpful than hateful in my experience; really good for figuring out new words, and for memorizing them. Plus, they're a beautiful part of the language. They take patience, but they're the aspect that makes Japanese most enjoyable for me, and their irregularities aren't that bad most of the time (usually one reading is used for ~95% of all words with that character).