To be visible the beam must either give off light itself, or excite the medium it is traveling through to give off light. In an atmosphere that would be the air. If the particles were dropping off energy in the air it could heat the air to glowing: this would be flames. If the particles very rapidly heated the air that would be analogous to lightning, and the rapid movement of air would be analogous to thunder.
The sudden increase in pressure and temperature from lightning produces
rapid expansion of the air within and surrounding the path of a
lightning strike. In turn, this expansion of air creates a sonic shock
wave, often referred to as a "thunderclap" or "peal of thunder".
As regards being visible in space, radiation to make them visible would have to come from the particle themselves. If you had enough mass of particles, perhaps in the process of their acceleration they would emit black body radiation according to their temperature.
A bolt of particle radiation would start out appearing white and then become redder as its constituent particles cooler during their journey. In a vacuum when the bolt became invisible because the particles had cooled they would have not lost any of their destructive power. You would need a metric boatload of particles for their glow to be visible in space. Your particle beam will converge on a superpowered shotgun shooting sand. More likely the particles would be less numerous and invisible.
If you used radon as your particle (as suggested below) or added some other intrinsically radioactive element you could follow the path of your beam if you had a device which could "see" in the frequencies emitted; alpha particles if you use radon or you could dope your ray with cobalt or some other gamma ray emitter. Making these radioactive elements should not be tough since your particle beam presumably works like a cyclotron.
Particle radiation can traverse matter in its path, like the atmosphere, or a body. Or your target. The place where most energy is dropped off is determined by the mass of the particle (proton? carbon ion? radon ion?) and the charge of the particle which mediates much of the interaction between particle and medium/. That place is called the Bragg peak. To get your shot thru the atmosphere but stopping in your target means you will need to aim in 3 dimensions: both the 2 dimensions of your plane of view and also the distance to the target where you want your particle to drop its energy. You will need to take into consideration what you are shooting thru. It is not outrageous to shoot at a target deep in the water or underground if you can make the particles energetic enough to traverse those media.
More on those matters at this question:
What subatomic particle is best for a particle accelerator gun?
Not to Bragg but here is a short fiction I wrote where a guy has a particle beam.