The interplay of spin-orbit coupling and strong electronic correlations is studied for the single-layer and the bilayer compound of the strontium ruthenate Ruddlesden-Popper series by a combination of first-principles band-structure theory with mean-field rotationally invariant slave bosons. At equilibrium strongly renormalized (spin-orbit-split) quasiparticle bands are traced and a thorough description of the low-energy regime for the nearly ferromagnetic bilayer system in accordance with experimental data is presented. The metamagnetic response of Sr3Ru2O7 in finite magnetic field H is verified and a detailed analysis of the underlying correlated electronic structure provided. Intriguing multiorbital physics on both local and itinerant level, such as competing paramagnetic and diamagnetic contributions, is observed with important differences depending on the magnetic-field angle. with the crystallographic c axis.