It takes a week to learn the system and to know the existence of at least all the hiragana characters and memorize the sound of some of them.

It takes two weeks to know know the existence of all the kana characters (including katakana), to memorize the sound of enough of them to read some words, and to write some of them.

After a month you should have easily memorized the sound of all of them (maybe a rare one like ム slips by occasionally), be able to write most of them, and be able to read (albeit slowly) anything written in kana.