To run (Like physically running) → 뛰다 (conjugation 뛰어, 뛰어요, 뛰어서, 뛰었어요)
To walk → 걷다 (걸어, 걸어요, 걸어서, 걸었어요)
To care about something → 신경 쓰다
To care about someone → There're a few phrases that carry the different nuance, I don't know which I should pick~
To cheat (On a test) → 컨닝하다(Cunning - Konglish)
To cheat (On a bf\gf) → 양다리 걸치다(informal and more common, 양다리 걸쳤어, 걸쳐서), 바람 피다
To kick (Like with your legs. Injury XD) → 차다, 걷어차다(stronger/informal)
To punch\slap\hit → 주먹으로 치다 / 손바닥으로 때리다 / 치다 (주먹으로 쳤어, 쳐서 / 때려, 때렸어)
"To become good\more good at something" → 점점 (더) 잘 하다(We don't use the passive form in this context)
Adjectives:
~smart (for people) 똑똑한
~stupid\dumb 멍청한 / 어리바리한 / 모자라는(informal)
~lazy 게으른
~horrible 무서운
~gross 징그러운(most common)
~evil 사악한