Go is used with activities and sports that end in -ing. E.g. go swimming
Do is used with recreational activities and with individual, non-team sports or sports in which a ball is not used,for example: do a crossword puzzle, do athletics, do karate.
Play is generally used with team sports and those sports that need a ball. Also, those activities in which two people or teams compete against each other: play football, play poker, play chess.
