Moderate-resolution (150-300 km s(-1)) spectra of the H beta-[O III] lambda 5007 region have been obtained for 41 radio-loud quasars in order to investigate relationships among their broad- and narrow-line profiles, radio structures, and X-ray properties. Comparable spectra from the literature have been included to form a data set of 60 radio-loud quasars with similar numbers of core- and lobe-dominated quasars. A variety of statistical analyses have been performed, including rank correlation tests and principal-component analysis, which identify several strong, related trends. The dominant source of variation in the H beta line profile parameters is related to radio core dominance, which measures the orientation of the central engine in the relativistic-beaming model (both the traditional R and a new core-dominance parameter R(v) are used). As core dominance increases, the equivalent widths EW(H beta) and EW(5007) decrease, H beta wings (the red in particular) weaken and H beta becomes more symmetric, and Fe II/[O III] lambda 5007 increases. The above changes in H beta are interpreted as the spectrum-to-spectrum variation of a redshifted, very broad component (FWHM similar to 8000-9000 km s(-1)). Secondary variations in H beta appear to involve a narrower component (FWHM similar to 2000 km s(-1)), with a strength unrelated to the core dominance or luminosity but strongly correlated with the strength of[O III] lambda 5007. Another set of significantly correlated parameters involves the [O III] lambda 5007 profile, continuum luminosity, and spectral shape: [O III] lambda 5007 is broader when the optical-to-X-ray spectral index alpha(ox) (F-v alpha v(-alpha)ox) and the optical luminosity are larger; the narrow-line (NLR) component of H beta also gets stronger relative to [ O III] lambda 5007, and there is a component of variation in EW(H beta) and EW(5007) in the sense of a Baldwin effect. These results, as well as some noteworthy noncorrelations, are briefly discussed in terms of underlying physical causes and other results from the literature. A quantitative comparison with other classes of active galactic nuclei is deferred to a forthcoming paper.