What is the equivalent of "alphabetical order" in Japanese? I know there's , but what about at the consonant level? Also, are there any common mnemonics used by Japanese children to remember these?

Hiragana syllables are always schematised in a given order which is this one (from right to left, as you may already know):

* * *

**HIRAGANA** !enter image description here

* * *

**KATAKANA** !enter image description here

* * *

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.
After you master it, you can start learning Kanji to substitute the "hiragana word" with the appropriate Kanji. I remember I used to write them, some examples are:

* ( _kindness/gentleness_ which is );
* ( _desk/table_ which is );
* ( _mountain_ which is ).