  We present the discovery and characterization of two sub-Neptunes in close
orbits, as well as a tentative outer planet of a similar size, orbiting
TOI-1260 - a low metallicity K6V dwarf star. Photometry from TESS yields radii
of $R_{\rm b} = 2.33 \pm 0.10$ $R_{\oplus}$ and $R_{\rm c} = 2.82 \pm 0.15$
$R_{\oplus}$, and periods of 3.13 and 7.49 days for TOI-1260b and TOI-1260c,
respectively. We combined the TESS data with a series of ground-based follow-up
observations to characterize the planetary system. From HARPS-N high-precision
radial velocities we obtain $M_{\rm b} = 8.61 _{ - 1.46 } ^ { + 1.36 }$
$M_{\oplus}$ and $M_{\rm c} = 11.84 _{ - 3.23 } ^ { + 3.38 }$ $M_{\oplus}$. The
star is moderately active with a complex activity pattern, which necessitated
the use of Gaussian process regression for both the light curve detrending and
the radial velocity modelling, in the latter case guided by suitable activity
indicators. We successfully disentangle the stellar-induced signal from the
planetary signals, underlining the importance and usefulness of the Gaussian
Process approach. We test the system's stability against atmospheric
photoevaporation and find that the TOI-1260 planets are classic examples of the
structure and composition ambiguity typical for the $2-3$ $R_{\oplus}$ range.
