The relativistic coupled cluster method is applied to the transition energies (ionization potentials, excitation energies, electron affinities) of lanthanum, actinium, and eka-actinium (element 121) in several ionization states. Good agreement with available experimental data is obtained. Large relativistic effects are observed for all three atoms, with transition energies changing by up to 2.5 eV even for the relatively light La. The first ionization potential of La is reduced from 7.135 to 5.582 eV (the experimental value is 5.577 eV). In Ac, the 7s shell is occupied before 6d, yielding 7s and 7s2 ground states for Ac2+ and Ac+, respectively, instead of the 5d and 5d2 states for the La ions. The ground state of E121 is 8s28p, to be compared with the ns2(n-1)d configurations of the lighter group-3 elements. The Breit effect on transition energies is significant (0.06-0.1 eV) for states including f orbitals and small (below 0.02 eV) for others. The contribution of virtual i orbitals (l=6) shows similar behavior. The electron affinities are predicted at 0.33 eV for La, 0.35 eV for Ac, and 0.57 eV for E121.