We present the discovery and characterization of two transiting exoplanets orbiting the metal-poor, solar-type star TOI-5788. From our analysis of six TESS sectors and a dedicated CHEOPS programme, we identify an inner planet (TOI-5788 b; $P = 6.340758\pm 0.000030\, d$) with radius $1.528\pm 0.075\, \mathrm{R_{\oplus }}$ and an outer planet (TOI-5788 c; $P = 16.213362\pm 0.000026\, d$) with radius $2.272\pm 0.039\, \mathrm{R_{\oplus }}$. We obtain 125 radial-velocity spectra from HARPS-N and constrain the masses of TOI-5788 b and c as $3.72\pm 0.94\, \mathrm{M_{\oplus }}$ and $6.4\pm 1.2\, \mathrm{M_{\oplus }}$, respectively. Although dynamical analyses indicate that a third planet could exist in a stable orbit between 8 and 14 d, we find no evidence of additional planets. Since the TOI-5788 system is one of the few systems with planets straddling the radius gap, and noting that there are even fewer such systems around metal-poor stars, it is a promising system to constrain planet formation theories. We therefore model the interior structures of both planets. We find that TOI-5788 b is consistent with being a rocky planet with almost no envelope, or having an atmosphere of a high mean molecular weight. We find that TOI-5788 c is consistent with both gas-dwarf and water-world hypotheses of mini-Neptune formation. We model the atmospheric evolution history of both planets. While both scenarios are consistent with the atmospheric evolution of TOI-5788 c, the gas-dwarf model is marginally preferred. The results of the atmospheric evolution analysis are not strongly dependent on stellar evolution. This makes the system a promising target to test internal structure and atmospheric evolution models.