We observed a sample of 20 representative Herbig Ae/Be stars and 5 A-type debris discs with PACS onboard Herschel, as part of the GAS in Protoplanetary Systems (GASPS) project. The observations were done in spectroscopic mode, and cover the far-infrared lines of [O i], [C ii], CO, CH, H2O, and OH. We have a [O i] 63 μm detection rate of 100% for the Herbig Ae/Be and 0% for the debris discs. The [O i] 145 μm line is only detected in 25% and CO J = 18–17 in 45% (and fewer cases for higher J transitions) of the Herbig Ae/Be stars, while for [C ii] 157 μm, we often find spatially variable background contamination. We show the first detection of water in a Herbig Ae disc, HD 163296, which has a settled disc. Hydroxyl is detected as well in this disc. First seen in HD 100546, CH emission is now detected for the second time in a Herbig Ae star, HD 97048. We report fluxes for each line and use the observations as line diagnostics of the gas properties. Furthermore, we look for correlations between the strength of the emission lines and either the stellar or disc parameters, such as stellar luminosity, ultraviolet and X-ray flux, accretion rate, polycyclic aromatic hydrocarbon (PAH) band strength, and flaring. We find that the stellar ultraviolet flux is the dominant excitation mechanism of [O i] 63 μm, with the highest line fluxes being found in objects with a large amount of flaring and among the largest PAH strengths. Neither the amount of accretion nor the X-ray luminosity has an influence on the line strength. We find correlations between the line flux of [O i] 63 μm and [O i] 145 μm, CO J = 18–17 and [O i] 6300 A, and between the continuum flux at 63 μm and at 1.3 mm, while we find weak correlations between the line flux of [O i] 63 μm and the PAH luminosity, the line flux of CO J = 3–2, the continuum flux at 63 μm, the stellar effective temperature, and the Brγ luminosity. Finally, we use a combination of the [O i] 63 μm and 12CO J = 2–1 line fluxes to obtain order of magnitude estimates of the disc gas masses, in agreement with the values that we find from detailed modelling of two Herbig Ae/Be stars, HD 163296 and HD 169142.