The paper presents the possibility of recovering metals from printed circuit boards (PCBs) of spent mobile phones using the hydrometallurgical method. Two-stage leaching of Cu(II), Fe(III), Sn(IV), Zn(II), Ni(II) and Pb(II) with H2SO4 (2 and 5 M) and HNO3 (2 M) with the addition of H2O2 (10 and 30%) and O3 (9 or 15 g/h) was conducted at various process conditions (temperature—313, 333 and 353 K, time—60, 120, 240, 300 min, type and concentration of leaching agent, type and concentration of oxidant, solid–liquid ratio (S/L)), allowing for a high or total metals leaching rate. The use of two leaching stages allows for the preservation of selectivity, separation and recovery of metals: in the first stage of Fe(III), Sn(IV) and in the second stage of the remaining tested metal ions, i.e., Cu(II), Zn(II), Ni(II) and Pb(II). Removing Fe from the tested PCBs’ material at the beginning of the process eliminates the need to use magnetic methods, the purpose of which is to separate magnetic metal particles (ferrous) from non-magnetic (non-ferrous) particles; these procedures involve high operating costs. Since the leaching of Cu(II) ions with sulfuric(VI) acid practically does not occur (less than 1%), this allows for almost complete transfer of these ions into the solution in the second stage of leaching. Moreover, to speed up the process and not generate too many waste solutions, oxidants in the form of hydrogen peroxide and ozone were used. The best degree of leaching of all tested metal ions was obtained when 2 M sulfuric(VI) acid at 353 K was used in the 1st research stage, and 2 M nitric(V) acid and 9 g/h O3 at 298 K in the 2nd stage of leaching, which allowed it to be totally leached 100% of Fe(III), Cu(II), Sn(IV), Zn(II), Ni(II) and 90% Pb(II).