A specific class of multiferroic superlattices R2CoMnO6/La2CoMnO6 (R = Ce, Pr, Nd, Pm, Sm, Gd, Tb, Dy, Ho, Er, Tm), which displayed observable electric polarizations and considerable magnetization, were investigated based on density functional theory. The multiferroic behavior was induced by both of the a(-)a(-)c(+) Glazer rotation patterns of BO6 (CoO6 and MnO6) octahedra and ferromagnetic coupling in the magnetic ordered superlattices. In addition, the ferroelectric and ferromagnetic properties of R2CoMnO6/La2CoMnO6 superlattices can be tuned by chemical pressure and hydrostatic pressure, with the former being more effective in tuning magnetoelectric properties than the latter. For chemical pressure, the incorporation of lanthanide ions promoted an increase of BO6 octahedral tilting, reflected by the sharp decrease of Co-O3-Mn bond angles in the R-layer along the c axis. By contrast, the hydrostatic pressure acts on all three directions of the superlattice so that the change in Co-O-Mn bond angles is relatively small, therefore the octahedral distortion is much smaller than that caused by chemical pressure. Consequently, the electric polarization and magnetization changed more slowly. Our first-principles simulations proposed a series of rational multiferroic superlattices with tunable ferromagnetism and ferroelectricity by chemical and hydrostatic pressures, with expectation to be applied as novel spintronic materials.