I know there's あいうえお, but what about at the consonant level? Also, are there any common mnemonics used by Japanese children to remember these?
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Hiragana syllables are always schematised in a given order which is this one (from right to left, as you may already know):
After a while I started Japanese, I also found this video, the Japanese Alphabet song (only for hiragana), that can be interesting for absolute beginners.
One of the best ways to remember Hiragana (and Katakana as well), in my opinion, is to write vocabulary words using Hiragana. It might seem banal, but it works.
The order used today pretty much everywhere is called Gojūon, and this is what Alenanno has described, but there's also a traditional order, with its own song, called Iroha. This order is actually based on the poem (instead of the other way around), and interestingly enough, it is possible because the poem managed to include every letter of the traditional alphabet exactly once. It was less useful than the Gojūon, because you needed to remember the song to use it, and over time the kana have lost two of the letters in Iroha (ゑ and ゐ) and gained a new one (ん).
The order is
あ か さ た な は ま や ら わ
If you have a Japanese cell phone, you can use the keypad to check the order, which runs from the 1 to 0 keys. If you have an iPhone, you can activate a Ten Key Japanese keyboard in the Keyboard settings, which is in the same order.
You can also listen to this cheesy Japanese hip-hop song by Kreva to hammer it into your head. The chorus is: