The breakdown is fairly simple as a general rule. As with many things in English, there are oddities, but following this structure will see a person through most cases.
play -> for games, instruments (eg, play soccer, play chess, play capoeira**, play piano, play guitar)
go -> ~ing sports/extended actions (eg, go swimming, go bungie jumping, go dancing)
do -> crafts, martial arts, performances (eg, do ballet, do karate, do crochet, do capoeira**)
A person also does piano lessons (because in this context, piano is used as an adjective modifying lessons). However, it would be incorrect to say "I go playing piano." (Not incorrect, but has a different meaning than it would generally hold in the discussed context).
**Capoeira - This can take "do" if a person is unaware that Capoeira is a sport before it is a martial art. However, it is more correct (and technically only correct) to say one "plays capoeira". - It is Brasil's national sport.
There are other oddities like these, but they are best learned as encountered rather than trying to memorize them right off. By using the general rule, it will be easier to figure out the exceptions as they arise.