I will describe the most important experimental data which led to creation of quantum mechanics, in the chronologic order of their explanations, not the order of experiments.
The idea of quanta was initially motivated by the theory of black body radiation. Plank derived his formula generalizing other formulas which came from experiments. To explain his empiric formula he had to invent the energy quanta (1900).
Another phenomenon which was unexplained by classical physics was photoelectric effect. (This is probably the simplest experiment to explain to beginning students. Understanding this work of Einstein requires almost no background in physics or mathematics, unlike other work mentioned here). Einstein explained what we observe in photoelectric effect by using Plank's idea of quanta, and extending it to electromagnetic radiation (1905).
Another experimental data which defied classical explanation were spectra of atoms. Bohr used the idea of quanta and explained the Balmer lines of hydrogen. Balmer discovered his empiric formula for the spectral lines in 1885 and Bohr "explained" it in 1913.
These are three kinds of experimental data from which quantum mechanic was born, historically. All three named physicists (Planck, Einstein and Bohr) were eventually awarded Nobel prizes for these discoveries, but it took some time, until 1930s before quantum mechanics obtained its modern form (Heisenberg, Born, Jordan, Dirac, Schrodinger and von Neumann).
Slit experiments played a role later than the three pieces of experimental data mentioned above. There was nothing unusual for physicists, since 19th century in the slit experiments with light. But discovery of electron diffraction in 1924 confirmed quantum mechanics.
Even more important was the Stern-Gerlach experiment (1922) which led to the discovery of spin. This eventually led to the explanation of the most important experimental fact of all mentioned: the Periodic Table.
Added on 3.4.2021. An outstanding exposition of the history of spectral lines inis the article by S. Sternberg, A history of 19th century spectroscopy (one of the very best articles on history of science that I know). It is published as Appendix F to his book "Group theory and physics".
On Stern-Gerlach (and Einstein-Podolsky-Rosen) there is a nice book by Jim Baggott, "The meaning of quantum theory".
Unfortunately I don't know any good exposition of Planck's discovery.