-으려고 (and -으러) * 선물하려고 케이크를 살 거예요 -려고 is probably more common in the past and present tense, but it can also be used for future plans like this. It can be rewritten as 선물하기 위해 ... or 선물용으로 ... but to me 선물하려고 sounds better than these. * 기억하려고 메모하세요 does sound unnatural. 기억하도록 메모하세요 or 기억할 수 있게 메모하세요 would be better. It still sounds odd however because the only reason for 메모하다 is to remember it. (Naver Dictionary: 다른 사람에게 말을 전하거나 자신의 기억을 돕기 위하여 짤막하게 글로 남기다.) * In order to graduate, I will study hard => 졸업하기 위해 [졸업을 위해, 졸업하려고] 열심히 공부할 것이다. * In order to make it on time, please walk faster => 시간에 맞추게 [맞도록] 더 빨리 걸어. (We would more often say 시간(에) 늦지 않게 더 빨리 가자 [걸어, 걷자]. * In order to be successful, you have to work harder => 성공하기 위해선 더 열심히 (일)해야 한다. -(으)러 indicates an immediate purpose for actions related to coming, going, or visiting a place. So these expressions are related but have graded connotations and sometimes different connotations. 1. -(으)러: immediate and concrete intention (for making a move). 2. -(으)려고: a fairly practical and easily achievable goal (for general actions). Has more restriction to usage. 3. -기 위해(서): a more serious or long term goal. * -게 can substitute these sometimes, especially for more casual situations. * -(으)려고 is special in that it's usually explains personal (or a group's) intentions that already exist (so it can't be used imperatively).

you also get another choice of 느라고

I'm on my phone so I can't write along answer, but (으)러 is different from (으)려고 in that its use with verbs is more limited. (으)러 is only used with directional verbs. (I went to the bank to deposit money. I went downstairs to turn the oven on. I came to Korea to study business)