The hyperphysics website is brilliant for beginners to study almost any field of physics. I recommend you read the specifics of what you want to know there.
Quantum mechanics was first indicated by the Schrodinger Equation, which solved the ‘Ultraviolet Catastrophe’ issue posed by applying classical formulae to the black body problem, and Einstein’s work on the Photoelectric Effect. The Schrodinger Equation treats light as being in distinct packets (quanta) rather than being a wave form as described by Maxwell’s electromagnetic equations. This idea was also important in Einstein’s explanation of the photoelectric effect.
The other major origin of quantum mechanics was the De Broglie hypothesis, which suggested that electrons are in the form of matter waves in order to solve the stability problems with the Bohr atomic model. So we have light being treated as particles (photons) and electrons being treated as waves. This soon led to the conclusion that particles and waves are just different expressions of the same entity, and either nature could be measured depending on the experiment performed. Experiments such as the double-slit experiment and the Bragg diffraction experiment confirmed this idea.
Further mathematical fiddling by some of the most brilliant physicists of the 20th century led to further implications of QM such as the Pauli Exclusion Principle, which prevents multiple particles possessing the same quantum numbers (such as mass, spin, charge and location in space and time). Heisenberg’s Uncertainty Principle is another important outcome, as it describes the limits on the knowledge it is possible to have about a particle. The more accurately we measure its position, the less accurately we can measure its velocity (and vice versa). This shows from a mathematical derivation that particles are not points in space, but are probability waves smeared through space.
QM is one of the most important and most mind bending scientific theories we have. I have not done it justice, but hopefully I’ve given you a few more leads to research.