The possibility of detection of 5.5 MeV and 14.4 keV solar axions by observing axion-induced nuclear and atomic transitions is investigated. The presence of nuclear transitions between spin orbit partners can be manifested by the subsequent deexcitation via gamma ray emissions. The transition rates can also be studied in the context of radiative axion absorption by a nucleus. The elementary interaction is obtained in the context of the axion-quark couplings predicted by existing axion models. Then, these couplings will be transformed to the nucleon level utilizing reasonable existing models, which lead to effective transition operators. Using these operators, we calculate the needed nuclear matrix elements employing wave functions obtained in the context of the nuclear shell model. With these ingredients, we discuss possibilities of experimental observation of the axion-induced nuclear gamma rays. In the second part, we will examine the axion-induced production of X-rays (axion-photon conversion) or ionization from deeply bound electron orbits. In this case, the axion electron coupling is predicted by existing axion models; no renormalization is needed. The experimental signal is the observation of directly produced electrons and/or the emission of hard X-rays and Auger electrons, following the deexcitation of the final atom. Critical discussion is made on the experimental feasibility of detecting the solar axions by using multiton scale NaI detectors.