Like, as (used to express similarity or manner: 'in the way that...', 'as if').
彼はライオンのように勇敢だ。
He is brave like a lion.
子どもが喜ぶように飾り付けをした。
I decorated it so that the children would be pleased (in a way children like).
音楽を聴くように本を読む。
I read a book as if listening to music (i.e., casually/with a relaxed rhythm).
So that; in order that (expressing purpose or a desired outcome: 'so that...').
忘れないようにメモしておいてください。
Please make a note so you don't forget.
皆が来られるように時間を早めます。
I'll move the time earlier so that everyone can come.
健康のために運動するようにしている。
I try to exercise so that I stay healthy.
Be sure to; make sure that (used to give instructions or strong advice: 'be sure to...').
時間通りに出発するように。
Make sure to depart on time.
鍵を忘れないように！
Be sure not to forget your keys!
I hope (that); may (used at the end of sentences after the -masu form to express wishes or prayers).
試験がうまくいきますように。
I hope the exam goes well.
病気が早く治りますように。
May your illness be cured quickly.