Over the past decades, a great deal of interest has been given to quasi-2D materials, with a large experimental effort being invested to unveil their electronic correlation properties. Some novel compounds exhibit complex phase diagrams, with the emergence of charge-density wave (CDW) or superconductivity, whose nature is still not clear, although being associated with strong electron-phonon (el-ph) interactions. These findings have raised issues about the relevance of the el-ph coupling for correlated materials, rather than just the electron-electron one. Given this motivation, here we present an overview on recent results about the role of the el-ph interaction to these compounds -- from transition-metal dichalcogenides to the twisted bilayer graphene. We add our contribution to this discussion, from a theoretical point of view, by developing and performing state-of-the-art quantum Monte Carlo simulations for the Hubbard-Holstein Hamiltonian, the "standard model" for this kind of system. Starting with simple geometries, such as the square lattice, we study the competition and interplay between antiferromagnetism (AFM) and CDW, establishing rich phase diagrams. Interestingly, in the region between AFM and CDW phases, we have found an enhancement of superconductivity, with favored unconventional pairing. Similarly, we also determine semimetal-to-insulator quantum critical points in the honeycomb lattice, the same geometry as the graphene. Dealing with interacting Dirac fermions leads to quite different results: the critical points are very susceptible to changes in the phonon frequencies, drastically affecting the phase diagram. On the other hand, we show that the el-ph interaction does not change the AFM universality class of the phase transitions. Indeed, understanding how to control and manipulate these many-body states remains a challenge for experimental physicists, and it is the bottleneck for eventual applications on semiconducting devices. As a step towards this end, our theoretical findings may shed light on the emergence of correlated properties in these quasi-2D materials.