The ATLAS Experiment measures the properties of particles created in the high-energy proton-proton collisions delivered by the Large Hadron Collider at CERN. The Standard Model of particle physics is our best description of the subatomic world - this well-studied theory provides accurate and precise descriptions of the fundamental particle properties and their interactions. Multiple areas of the Standard Model make predictions that are more precise than the corresponding experimental measurements - in some cases, increased experimental precision could suggest significant discrepancies between the Standard Model prediction and experimental measurement, potentially hinting at new physics to elucidate. One such area of tension between prediction and experiment is in the weak sector, whose force is mediated by W and Z bosons. The current best predictions based on the Standard Model do an inadequate job of precisely predicting one important observable of W and Z bosons: their transverse momentum, p_T. Precision measurements of properties like the p_T of W and Z bosons improve our knowledge of the weak sector, and are vital stepping stones to critical measurements like the mass of the W boson, whose most recent reported measurement is inconsistent with the Standard Model prediction. In this thesis, I explain my work using ATLAS data to make high-precision measurements of the p_T of W and Z bosons using the decay channels W --> l nu and Z --> l l (l = e, mu) at centre-of-mass energies 13 and 5 TeV using a special low-pileup dataset. I show that I have helped reduce the systematic uncertainties to percent-level precision and that statistical uncertainties are dominant, which demonstrates that more low-pileup data should be taken in order to further reduce the total uncertainty to eventually help resolve Standard Model weak-sector discrepancies like that of the W boson mass. I also detail my work to improve the way that electrons are identified by the ATLAS detector using a technique called W tag-and-probe. In particular, I validated the use of a new trigger designed specifically for electron identification with the W tag-and-probe technique.