First of all, welcome to the site. What quantum mechanics tells us is not that processes are random; but that there is a "fundamental" (at least from the QM point of view) impossibility of having certain informations (for example the trajectory of a particle) with perfect precision. That means that the maximal informations we can have of the state of a system is of probabilistic nature, id est we are only able to say "the outcome $a$ of a measure about the quantity $A$ have a $x\%$ probability of being measured".
That does not mean, however, that the processes happen randomly, they are governed anyways by physical laws; nevertheless these same laws forbid a completely precise "classical" description of the system. So I may say that randomness, in this particular context, is not a good word, also because it may be misleading since the study of random processes is an important branch of mathematics and physics (but not strictly related to basic quantum mechanics).
Related to the OP comment above: those processes are not random from a QM point of view. Simply QM introduces, as I said, the concept of probability of a measure as opposed to exact knowledge. Take the specific example of the photon interacting with an electron. What we can say, very roughly speaking, is that there is only a certain probability that the photon would interact with the electron (and that probability depends on the state of the system photon+electron); and that this causes the electron to absorb the energy of the photon. After that, there is a certain probability that the electron would lose energy and emit a photon, perhaps of different energy. That does not mean that the events are random, they are dictated by the laws of quantum electrodynamics; it simply means that the maximal information that the QM description can give is this, i.e. which is the probability of a certain event.
It is a different type of prediction, that changes radically the point of view from the classical one: if in classical mechanics we would have predicted, given the initial conditions, exactly if the photon and the electron (seen as classical particles) would collide (as in a billiard); on the contrary in quantum mechanics we do not know a priori if the collision happens, we know only which is the probability of the collision. However there is nothing random about that probability, it is determined exactly by the quantum theory (as the trajectory was in classical mechanics).