Very accurate ab initio electronic + spin-orbit calculations of the lowest-lying states of the Ag atom and Ag+ cation have been performed through the CASSCF + ACPF + EPCISO method, using the Stuttgart small-core (19 active electrons) relativistic effective core potential (RECP) as well as its associated D-2 spin-orbit effective potential. An ad hoc spin-orbit P-symmetry pseudopotential for the P-2 state adapted to this 19-e RECP and basis set was extracted. The Stuttgart basis set was augmented to a large valence Gaussian basis set (8s8p7d3f3g/6s6p4d3f3g) in order to reproduce at best the experimental S-2-D-2 and S-2-P-2 transition energies as well as the ionization potential (IP) of Ag, which play a crucial role for the accurate description of the spectroscopy in silver-containing molecular systems. A detailed discussion on the multiple schemes used to deal with the differential d(10) vs d(9) electronic correlation for these two excited states is given. The role of the 4s and 4p (core) shells on the S-2-D-2 and S-2-P-2 transition energies and the IP is carefully studied and discussed. The core-core correlation is found to play a minor role while an insufficient treatment of the core-valence electronic correlation is responsible for the main differential d(10) vs d(9) correlation energy error between the S-2-D-2 and S-2-P-2 transition energies. For the neutral atom, the D-2(5/2)-D-2(3/2) and P-2(3/2)-P-2(1/2) splittings are in excellent agreement with the experimental ones. However, the relative calculated energetic ordering for the D-2(5/2),D-2(3/2),P-2(3/2), and P-2(1/2) fine structure components is critically dependent on the J-averaged purely electronic ACPF P-2 and D-2 energies of the parent states. The D-3 fine-structure splitting for the ion is also found in good agreement with the experiment.