The curriculum guidelines for grade one (see 言語事項 section イ) only state that children should be able to read and write hiragana and katakana, and use words that are written in katakana in sentences (e.g. know to write ペン not ぺん), and to read and to start to use the level-appropriate kanji.
As I understand it, instead of memorising individual readings, the focus is usually on being able to read/read aloud level-appropriate materials, or supply the correct kanji in the context of a given word in a sentence (e.g. use the right kanji for the right word when writing assignments). There's also some focus on stroke order and handwriting.
Materials supplied on the internet for practice, which are aimed at school children (or their parents), for example here, which has various levels of material, tend to use these sort of formats for kanji: writing practice (with characters to trace over), reading practice, more writing practice (give the kanji based on the furigana).
There are "official" readings (which you will see in sawa's second link in the comments on dotnetN00b's answer). They are expected to be known, in that words using those readings should be able to be read, but I've never seen any material that asks "what are the on-readings of this kanji" aimed at native speakers.
This page has guidance materials aimed at teachers using certain textbooks. For the first year version, in this case they start with hiragana, then it looks like they actually pick up a few kanji, like numerals, before getting into katakana. (Although to be entirely honest I only skimmed it, and I don't know how representative the provided plan is of the average class).
