We study the positivity properties of Ulrich bundles defined with respect to an ample and globally generated polarization. First we prove a generalization of a theorem by Lopez on the first Chern class. Then, under some additional assumptions on the polarization, we give a description of its augmented base locus, which consequently leads to a characterization of the V-bigness and of the ampleness of an Ulrich bundle in this setting. Finally we study the normal generation of an Ulrich bundle focusing on curves, on surfaces with $q = p_g = 0$ and on hypersurfaces of dimension 2 and 3