ために and ように: goals and desired states
Do not choose a purpose expression from the translation 'in order to' alone. Consider whether the goal is an intended action or a state to achieve or avoid.
Making an intended action the goal
日本へ留学するために、お金を貯めています。 means saving money to study in Japan. The intended action, 留学する, is the goal; 貯める describes the action taken to support it.
With a noun, add の, as in 留学のために. Since ため also has a causal use, check whether the later action serves the goal stated before ために. Do not automatically read every use as purpose.
Aiming for a result or ability
日本語の新聞が読めるように、漢字を勉強しています。 means studying kanji so that one can read a Japanese newspaper. The goal is the ability expressed by 読める, rather than simply the action 読む.
忘れないように、予定をメモします。 means noting the schedule so as not to forget it. A negative form can name a result to avoid. ように also works across different subjects, as in 後ろの人にも見えるように: so that the people at the back can see too.
Formulating the goal before choosing
An intention to take an exam differs from a goal of being able to understand instructions. First distinguish intended actions such as 受ける from states or abilities such as 分かる, 聞こえる, and 読める.
Do not automatically swap the expressions based on one verb in isolation. The same activity can be framed through different goals. When reading, separate the goal from the means used to pursue it.
Check: Complete 'I make a note so I do not forget': 忘れない＿＿、メモします。
Show the answer and why
The basic purpose expression is 忘れないように、メモします。 Not forgetting is the desired outcome, and making a note is the action taken toward it.