In this work, we develop a general theoretical scheme to study tracer-diffusion in mixtures of interacting colloidal particles where the influence of solvent-mediated hydrodynamic interactions is also considered. Based;on the many-body Smoluchowski diffusion equation, we derive in a first step an exact microscopic expression of the irreducible memory function (self-friction function) associated with the self-intermediate scattering function and with the mean squared displacement of a tagged particle. By applying a mode-coupling scheme (MCS) to the irreducible memory function, we obtain explicit expressions for the tracer-diffusion quantities in terms of partial static structure factors and hydrodynamic functions. The influence of hydrodynamic interactions (HI) is accounted for using a far-field expansion of the two-body hydrodynamic diffusivity tensors. For charge-stabilized colloids, this is a good approximation due to strong electrostatic repulsion between the particles. Various applications are discussed in order to illustrate the versatility of our mode-coupling scheme.