Since in theory matter can't be created or destroyed except through a nuclear reaction, what happens to an atom's subatomic particles when it is smashed?
Let us start from proton. If during any process a proton is produced it will get electron from somewhere around and will be converted in hydrogen atom which ultimately will find any other atom to attach with. There might be alpha particles one of the most stable nuclei, these particle will also get electrons from surrounding and convert into helium atoms. Heavy nuclear parts will adjust electron cloud accordingly and convert into corresponding atom according to the number of protons they have. There might be neutrons, neutrons just get absorbed by target nuclei, and hence the products of a smashed atom are consumed.
In experiments (eg the Relativistic Heavy Ion Collider) ionised atoms of Gold, Lead and others are collided, with some of the nuclear particles forming a quark-gluon plasma...in other words, being broken down into the constituent parts of protons and neutrons. This plasma condenses into a relatively random collection of particles. Some of these are short lived and decay into other particles. Ultimately, in a collider setting, most of them end up being absorbed by various detectors and/or shielding materials.