The processes that take place in the excited states of a trivalent Thulium (Tm) ion in an Yttrium Aluminum Garnet (YAG) crystal, being relevant to the use of this system for laser applications, have been the object of several studies. We have reexamined this system focusing our attention on the dynamics of Tm following its excitation in the H-3(sub 4) level. Under these conditions the system relaxes through a cross-relaxation process. H-3(sub 4) yields F-3(sub 4), H-3(sub 6) yields F-3(sub 4), whose rate depends upon both the concentration of the Tm ion and the temperature of the crystal. The excitation spectrum obtained by monitoring the 1.8 micron emission of Tm (due to the F-3(sub 4) yields H-3(sub 6) transition) indicates an increase in the contribution to this emission from the H-3(sub 4) level relative to the H-3(sub 5) level as the Tm concentration increases; this shows the increased role played by the H-3(sub 4) level in pumping the infrared emission. Correspondingly, the duration of the luminescence originating in the H-3(sub 4) level is shortened as the concentration of Tm increases. The concentration quenching of this lifetime can be fit to a model which assumes that the cross-relaxation is due to a dipole-dipole interaction; from this fit, the intrinsic Tm lifetime in the absence of cross relaxation can be derived. We have used this lifetime to calculate the rate of the cross-relaxation process. We have evaluated this rate as a function of the temperature and found it to be fastest at 77 K. We have also calculated the microscopic interaction parameters for the cross-relaxation process by using two independent experimental features: (1) the time evolution of the emission from the H-3(sub 4) level; and (2) the spectral overlap between the H-3(sub 4) yields F-3(sub 4) emission and the H-3(sub 6) yields F-3(sub 4) absorption. We have also considered the migration of excitation among the Tm ions in the F-3(sub 4) level and calculated the relevant microparameter by the use of the relevant spectral overlap. The data are consistent with the model in which the Tm ions, once excited into the H-3(sub 4) level decay by cross-relaxation to the F-3(sub 4), and then transfer rapidly their energy to other Tm ions.