play on the playground - 놀이터에서 놀다
play piano - 피아노를 치다 or 연주하다 (치다 directly means 'hit'. So with other instruments, you should choose other words. e.g. 'play flute' is '플루트를 불다' which directly means 'blow' flute. I think it makes you quite confused. So I recommend using '연주하다' which means 'play a song with instruments' and can be used generally )
play tennis - 테니스를 치다 or 하다 (As you hit the ball with a racket, '치다' is used. But in other sports like baseball, it isn't. So, you can just use '하다' in most sports.)
play checkers - 체커를 하다 (or '두다' meaning 'put' a piece)
play Juliet - 줄리엣을 연기하다
play someone - 누구를 속이다
(If it means 'play a prank on someone', you should say '장난치다')
play a song - 곡을 연주하다 (It means you play the song yourself. If you play a song with a phone, computer, or audio player, you should say 노래를/곡을 '틀다')
replay an event - Directly translated to '일이 재생되다', it sounds quite awkward. So can represent it like '계속 생각이 나다/떠오르다' meaning 'continuously the thoughts keep coming up'