Choose photons of high frequency if you want to ionise matter. These can be from X-ray machines or from ultra-violet light. They cannot be from mobile phones or microwaves. The sources set the frequency and the energy. Never mind how bright the source is, if each photon does not shift enough energy, you won’t get any ionising.
Some ionising radiations are so energetic that you have to look to another source: the nucleus. A nuclear change shifts much more energy than one that only involves the electrons of the atom, therefore any radiations emitted from the nucleus are ionising. They are energetic enough to ionise many molecules or atoms, so are damaging. The first three kinds found were called alpha, beta and gamma radiation.
There are some simple tests to decide which is which.
From these you can tell alpha and beta are electrically charged, but gamma is not. You can also tell that alpha is easiest to shield yourself from, and gamma hardest.
They are not all equally damaging.
Choose a short range radiations to do lots of damage over that short range. A long range radiation is much less damaging, but over a larger volume.
The longer range radiations have less chance of doing damage for each millimetre of matter that they go through. Each piece of damage strips the radiation of some energy. Put these two together to explain why short range radiations do lots of damage over that short range.