Despite over thirty years of research, the origin of high-temperature superconductivity remains unsolved. In these thirty years, the phase diagram for the rst-discovered high-temperature superconductors, the cuprates, has been found to be rather complex and exhibits many different phases such as antiferromagnetism, charge density waves, spin density waves, nematicity, the pseudogap, and of course, superconductivity. Furthermore, several structural instabilities can manifest that affect the stability of these phases. In the La-214 cuprates, for example, it is known the concomitant charge and spin orders (or stripe order) are stabilized by a low-temperature tetragonal structure. The stripe order coincides with a suppression of the superconducting critical temperature, leading the conclusion that these phases either compete or are intertwined. Since the stability of the low-temperature tetragonal structure, and therefore stripes, can be controlled by various dopants, the La-214 cuprates can be used to investigate how these orders intertwine. In this thesis, both striped and unstriped La-214 compounds have been investigated to understand the interplay of these various orders: superconductivity, stripes, and structure. In three distinct studies, using various charge transport techniques, the interplay between these orders is shown to lead to interesting and unexpected behavior. The first study reveals static charge order is in fact a fluctuating order pinned by the structure. The second study shows the two-dimensional nature of the superconductivity in the absence of stripe order, which is speculated to decouple CuO2 planes. Finally, the third study reveals the existence of a hidden order of Cooper pairs in the T=0 field-driven superconducting-normal-state transition when stripes are present. The culmination of these distinct studies lead to a better understanding of the physics of cuprates through the interplay of their various orders, and thus the general phase diagram of high-temperature superconductivity.