看 (kàn) can be used with a range of things that you can watch or read
看电视 (kàn diànshì) watch TV
看电影 (kàn diànyǐng) watch movies
看书 (kànshū) read a book
完 (wán) has a wide range of meaning from 'playing a game', to 'having fun' to 'going out'
完游戏 (wán yóuxì) to play a game
打 (dǎ) is commonly used in sense of to play a game or sport
打拍 (dǎ pái) to play cards
打麻将 (dǎ májiàng) to play Majhong
下 (xià) 'to play' is often used with board games
下跳棋 (xià tiàoqí) to play Chinese checkers
下象棋 (xià xiàngqí) to play Chinese chess