In the present paper, we apply the geometrical mechanism of diffusion in an a priori unstable Hamiltonian system [L. Chierchia and G. Gallavotti, Ann. l’I.H.P. Phys. théor. 60, 1–144 (1994)] with 3 + 1/2 degrees of freedom. This mechanism consists of combining iterations of the inner and outer dynamics associated with a Normally Hyperbolic Invariant Manifold (NHIM) to construct diffusing pseudo-orbits and subsequently apply shadowing results to prove the existence of diffusing orbits of the system. In addition to proving the existence of diffusion for a wide range of the parameters of the system, an important part of our study focuses on the search for Highways, a particular family of orbits of the outer map (the so-called scattering maps), whose existence is sufficient to ensure a very large drift of the action variables, with a diffusion time near them that agrees with the optimal estimates in the literature. Moreover, this optimal diffusion time is calculated, with an explicit calculation of the constants involved. All these properties are proved by analytical methods and, where necessary, supplemented by numerical calculations.