様(よう)にSo that, In order to, In such a way that

ように means 'in such a way' and expresses doing something to achieve a certain result, e.g. 忘れないようにメモした ('I took a memo so I wouldn't forget.'). ようになる, on the other hand, describes a situation changing to a certain state, without specifying the cause, e.g. 話せるようになった ('I became able to speak.').

The core meaning of ように is 'in such a way'. It's used to show an action done to achieve a result, e.g. 勉強できるように新しい本を買った ('I bought a new book so I can study.'). ようにする builds on this to express making a habit or effort, like 毎日、日本語を勉強するようにしている ('I try to study / make a habit of studying Japanese every day.').

Both structures express doing something in a certain way. れる・られる + ままに is used when someone acts as they're told, e.g. 言われるがままに 'just as told'. ように can express acting of one's own accord, e.g. 間に合うように早めに出た ('I left early so I could make it on time.').

The core meaning of ように is 'in such a way.' It shows doing something to achieve a result, e.g. 間に合うように早めに出かけた ('I left early so I could be on time.'). ようにいう builds on this and means 'to instruct someone to do something', e.g. 間に合うように言っておいた ('I told them to be on time.').

The core meaning of ように is 'in such a way'. In general usage, this conveys performing a certain action in order to achieve a stated result, e.g. 合格できるように毎日勉強した ('I studied every day so I could pass.'). ようにいのる (often written in kanji as 祈る 'pray') is one form of this and expresses a wish or hope, e.g. 合格できるように祈っているよ ('I'm hoping/praying you'll pass.').

The core meaning of ように is 'so that' or 'in order to', and it is commonly used in patterns like ようにする 'make an effort to (A)' or ようにいう 'ask someone to (A)'. When used together with the ～てほしい form, it expresses wanting someone else to do the action. For example, 忘れないようにする ('I'll try not to forget.') vs. 忘れないようにしてほしい ('I want you to try not to forget.').

如何 (read as いかん) is a formal term meaning 'how' or 'in what way', used mainly in formal or academic writing. It appears after nouns, often in the pattern 如何によって ('depending on'). 状況の如何で決まる = 'It will be decided depending on the situation.' ように is a more common, casual form, e.g. 忘れないようにメモした ('I took a memo so I wouldn't forget.').