why can the different types of ionizing radiation penetrate paper, aluminum and lead?

At the same time you could ask why are different types of radiation stopped by these barriers. Alphas (flying He nuclei) can be stopped by most anything, paper, skin. They deposit their energy in a short space. Betas go a little further because they are less massive, water will stop them. Gammas are massless and are very energetic. They are not spoken of as stopped, but as attenuated. Lead, being very dense as a barrier, can't stop gammas, only attenuate them. This means cut down the intensity of the beam of gammas. The importance of knowing the stopping ability is in shielding designs for nuclear reactors. A combination of shielding material is used to protect workers and the public in general.

Alpha particles (helium ions) are big and bulky. Just about anything will stop them. Beta particles (electrons) are much smaller and somewhat harder to stop. High energy gamma rays (EM radiation) are even harder still to stop.

It's a combination of factors--their mass, energy, and cross section (likelihood) to interact with those materials.