We use the Herschel/PACS spectrometer to study the global and spatially resolved far-infrared (FIR) fine-structure line emission in a sample of 52 galaxies that constitute the SHINING survey. These galaxies include star-forming, active-galactic nuclei (AGN), and luminous infrared galaxies (LIRGs). We find an increasing number of galaxies (and kiloparsec size regions within galaxies) with low line-to-FIR continuum ratios as a function of increasing FIR luminosity ($L\mathrm{FIR}$), dust infrared color, $L\mathrm{FIR}$ to molecular gas mass ratio ($L\mathrm{FIR}/M\mathrm{mol}$), and FIR surface brightness ($\Sigma \mathrm{FIR}$). The correlations between the [CII]/FIR or [OI]/FIR ratios with $\Sigma \mathrm{FIR}$ are remarkably tight ($\sim 0.3$ dex scatter over almost four orders of magnitude in $\Sigma \mathrm{FIR}$). We observe that galaxies with $L\mathrm{FIR}/M\mathrm{mol}\gtrsim 80L\odot M\odot -1$ and $\Sigma \mathrm{FIR}\gtrsim 1011$ $L\odot$ kpc$\mathrm{<mrow\; data-mjx-texclass="ORD">-2</mrow>}$ tend to have weak fine-structure line-to-FIR continuum ratios, and that LIRGs with infrared sizes $\gtrsim 1$ kpc have line-to-FIR ratios comparable to those observed in typical star-forming galaxies. We analyze the physical mechanisms driving these trends in Paper II (Herrera-Camus et al. 2018). The combined analysis of the [CII], [NII], and [OIII] lines reveals that the fraction of the [CII] line emission that arises from neutral gas increases from 60% to 90% in the most active star-forming regions and that the emission originating in the ionized gas is associated with low-ionization, diffuse gas rather than with dense gas in HII regions. Finally, we report the global and spatially resolved line fluxes of the SHINING galaxies to enable the comparison and planning of future local and high-$z$ studies.