Ultra-hot Jupiters are highly irradiated gas giants with equilibrium temperatures typically higher than 2000 K. Atmospheric studies of these planets have shown that their transmission spectra are rich in metal lines, with some of these metals being ionised due to the extreme temperatures. Here, we use two transit observations of WASP-76b obtained with the CARMENES spectrograph to study the atmosphere of this planet using high-resolution transmission spectroscopy. Taking advantage of the two channels and the coverage of the red and near-infrared wavelength ranges by CARMENES, we focus our analysis on the study of the Ca II infrared triplet (IRT) at 8500 Å and the He I triplet at 10 830 Å. We present the discovery of the Ca II IRT at 7σ in the atmosphere of WASP-76b using the cross-correlation technique, which is consistent with previous detections of the Ca II H&K lines in the same planet, and with the atmospheric studies of other ultra-hot Jupiters reported to date. The low mass density of the planet, and our calculations of the XUV (X-ray and EUV) irradiation received by the exoplanet, show that this planet is a potential candidate to have a He I evaporating envelope and, therefore, we performed further investigations focussed on this aspect. The transmission spectrum around the He I triplet shows a broad and red-shifted absorption signal in both transit observations. However, due to the strong telluric contamination around the He I lines and the relatively low signal-to-noise ratio of the observations, we are not able to unambiguously conclude if the absorption is due to the presence of helium in the atmosphere of WASP-76b, and we consider the result to be only an upper limit. Finally, we revisit the transmission spectrum around other lines such as Na I, Li I, Hα, and K I. The upper limits reported here for these lines are consistent with previous studies.