We examine the propagation of 2-dimensional relativistic jets through the stellar progenitor in the collapsar model for gamma-ray bursts. In agreement with previous studies, we find that relativistic jets are collimated by their passage through the stellar mantle. Interaction of these jets with the star and their own cocoons also causes mixing that sporadically decelerates the flow. We speculate that this mixing instability is chiefly responsible for the variable Lorentz factor needed in the internal shock model and for the complex light curves seen in many GRBs. In all cases studied, the jet is shocked deep inside the star following a brief period of adiabatic expansion. The jet that finally emerges from the star thus has a moderate Lorentz factor, modulated by mixing, and a very large internal energy. In a second series of calculations, we follow the escape of that sort of jet. Because of the large ratio of internal to kinetic energy in both the jet and its cocoon, the opening angle of the final jet is significantly greater than at breakout. A small amount of material emerges at large angles, but with a Lorentz factor still sufficiently large to make a weak GRB. This leads us to propose a "unified model" in which a variety of high energy transients, ranging from x-ray flashes to "classic" GRBs, may be seen depending upon the angle at which a standard collapsar is observed. We also speculate that the breakout of a relativistic jet and its collision with the stellar wind will produce a brief transient with properties similar to the class of "short-hard" GRBs. Implications of our calculations for GRB light curves, the luminosity-variability relation, and the GRB-supernova association are also discussed. (Abridged)

40 pages, 16 figures, To appear in vol. 586, ApJ, March 20, 2003