In British English, you can "do sport". In American English you can "play sports".
A mistake some speakers make is using the verb practice for sports:
*I love practicing sport. This should be: I love sport.
*I usually practice sport every evening. This should be: I usually do sport every evening.
Go is used with activities and sports that end in -ing. The verb go here implies that we go somewhere to practice this sport: go swimming.
Do is used with recreational activities and with individual, non-team sports or sports in which a ball is not used, like martial arts, for example: do a crossword puzzle, do athletics, do karate.
Play is generally used with team sports and those sports that need a ball or similar object (puck, disc, shuttlecock...). Also, those activities in which two people or teams compete against each other: play football, play poker, play chess.
You use do with two activities that end in -ing: do boxing and do body-building, because they don't imply moving along as the other activities ending in -ing.
Golf: if there is an idea of competition, you use the verb play. However, you can say go golfing if you do it for pleasure:
Tiger Woods plays golf.
We'll go golfing at the weekend.