We introduce SERRA, a suite of zoom-in high-resolution ($\sim 10\,\rm pc$) cosmological simulations including non-equilibrium chemistry and on-the-fly radiative transfer. The outputs are post-processed to derive galaxy UV+FIR continuum and emission line properties. Results are compared with available multi-wavelength data to constrain the physical properties (e.g., star formation rates, stellar/gas/dust mass, metallicity) of high-redshift $6 \lesssim z \lesssim 15$ galaxies. This flagship paper focuses on the $z=7.7$ sub-sample, including 202 galaxies with stellar mass $10^7 M_\odot \lesssim M_\star \lesssim 5\times 10^{10}M_\odot$, and specific star formation ranging from ${\rm sSFR} \sim 100\,{\rm Gyr}^{-1}$ in young, low-mass galaxies to $\sim 10\,{\rm Gyr}^{-1}$ for older, massive ones. At this redshift, SERRA galaxies are typically bursty, i.e. they are located above the Schmidt-Kennicutt relation by a factor $\kappa_s = 3.03^{+4.9}_{-1.8}$, consistent with recent findings for [OIII] and [CII] emitters at high-$z$. They also show relatively large ${\rm IRX} = L_{\rm FIR}/L_{\rm UV}$ values as a result of their compact/clumpy morphology effectively blocking the stellar UV luminosity. Note that this conclusion might be affected by insufficient spatial resolution at the molecular cloud level. We confirm that early galaxies lie on the standard $\rm [CII]-SFR$ relation; their observed $L_{\rm [OIII]}/L_{\rm [CII]} \simeq 1-10$ ratios can be reproduced by a part of the SERRA galaxies without the need of a top-heavy IMF and/or anomalous C/O abundances. [OI] line intensities are similar to local ones, making ALMA high-$z$ detections challenging but feasible ($\sim 6\,\rm hr$ for a SFR of $50\,M_\odot\,{\rm yr}^{-1}$).
Comment: 23 pages, 14 figures, accepted by MNRAS; part of the data used for this study is available at the website http://cosmology.sns.it/data_access.html