Being such a different language, Korean doesn't have a word closely matching the infinitive "to".
There are a few different verb forms that act like a "to". I will show the two major cases below.
1. indicate purpose or intention.
A. formal, explicit purpose ("in order to") = -기 위해(서) - this is not used very often in casual contexts.
B. casual "to" = -려고 (general), -러 (for words like "come", "go", etc), -게, -도록 ("for X to Y", "so as to").
- 피해를 줄이기 위해 예방 조치를 취하다 = take preventive measures to reduce damage.
- 공무원이 되려고 시험 준비를 하다 = prepare for the exam to become a civil servant.
- Alex를 찾으러 가야 해요 = We must go to find Alex. (-러 가다, -러 오다, -러 다니다 are idiomatic expressions)
- 교통이 원활해 지게/지도록 도로를 건설했다 = built a roadway (so as) to facilitate transportation.
- 톰을 영어를 가르치게/가르치도록 초빙하고 싶다 = I'd like to invite Tom to teach English. (This sounds awkward in Korean. We'd say 톰을 영어 강사로 초빙하고 싶다 or 톰에게 영어 수업/강의를 맡기고/부탁하고 싶다)
2. indicate resulting state or outcome - V1-아/어서 V2-되다/하다. V1-아/어서 V2-게 되다(서 may be dropped).
- 톰을 초빙해(서) 영어를 가르치게 하자 = Let's invite Tom to teach English.
- 그는 도박에 빠져(서) 결국 재산을 탕진했다 = He indulged in gambling, to lose all he had in the end.
There many other cases of "to" which correspond to various different verb forms in Korean.