We present observations and initial analysis from a Hubble Space Telescope (HST) Cycle 19 program using STIS to obtain the first co-spatial, UV–optical spectra of 10 Galactic planetary nebulae (PNs). Our primary objective was to measure the critical emission lines of carbon and nitrogen with unprecedented signal-to-noise ratio (S/N) and spatial resolution over the wavelength range 1150–10270 Å, with the ultimate goal of quantifying the production of these elements in low- and intermediate-mass stars. Our sample was selected from PNs with a near-solar metallicity, but spanning a broad range in N/O based on published ground-based and IUE spectra. This study, the first of a series, concentrates on the observations and emission-line measurements obtained by integrating along the entire spatial extent of the slit. We derived ionic and total elemental abundances for the seven PNs with the strongest UV line detections (IC 2165, IC 3568, NGC 2440, NGC 3242, NGC 5315, NGC 5882, and NGC 7662). We compare these new results with other recent studies of the nebulae and discuss the relative merits of deriving the total elemental abundances of C, N, and O using ionization correction factors (ICFs) versus summed abundances. For the seven PNs with the best UV line detections, we conclude that summed abundances from direct diagnostics of ions with measurable UV lines give the most accurate values for the total elemental abundances of C and N (although ICF abundances often produced good results for C). In some cases where significant discrepancies exist between our abundances and those from other studies, we show that the differences can often be attributed to their use of fluxes that are not co-spatial. Finally, we examined C/O and N/O versus O/H and He/H in well-observed Galactic, LMC, and SMC PNs and found that highly accurate abundances are essential for properly inferring elemental yields from their progenitor stars. Future papers will discuss photoionization modeling of our observations, of both the integrated spectra and spatial variations of the UV versus optical lines along the STIS slit lengths, which are unique to our observations. Based on observations with the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.