Standard cosmological models predict that the first astrophysical sources formed from a Universe filled with neutral hydrogen (HI) around one hundred million years after the Big Bang. The transition into Cosmic Dawn (CD) that seeded all the structures seen today can only be probed directly by the 21-cm line of neutral hydrogen. Redshifted by the Hubble expansion, HI signal during CD is expected to be visible in radio frequencies. Precisely characterized and carefully calibrated low-frequency instruments are necessary to measure the predicted ~10-200 mK brightness temperature of this cosmological signal against foregrounds. This dissertation focuses on improving the existing instrumental and analysis techniques for the Experiment to Detect the Global EoR Signature (EDGES) and building capabilities for future space-based 21-cm instruments, including the Farside Array for Radio Science Investigations of the Dark ages and Exoplanets (FARSIDE) concept.Frequency-dependent antenna beams of 21-cm instruments limit the removal of bright galactic foreground emission (~10^3 - 10^4K) from observations. Using three electromagnetic simulation packages, I modeled the EDGES low-band antenna, including the ground plane and soil, and quantified its variations as a function of frequency. I compared simulated observations to sky data and obtained absolute agreement within 4% and qualitatively similar spectral structures. I used the new open-source edges-analysis pipeline to carry out rigorous fits of the absorption feature on the same low-band data and lab calibration measurements as (Bowman et. al. 2018). Using a Bayesian framework, I tested a few calibration choices and found posteriors of the best-fit 21-cm model parameters well within the 1σ values reported in B18. To test for the ``global'' nature of the reported cosmic absorption feature, I performed a time-dependent analysis. Initial results from this analysis successfully retrieved physical estimates for the foregrounds and estimates of the cosmic signal consistent with previous findings. The array layout of FARSIDE, a NASA probe-class concept to place a radio interferometer on the lunar farside, is a four-arm spiral configuration consisting of 128 dual-polarized antennas with a spatial offset between the phase centers of its orthogonal polarizations. I modeled the impact of direction-dependent beams and phase offsets on simulated observations of all four Stokes parameter images of a model and quantified its effects on the two primary science cases: 21-cm cosmology and exoplanet studies.