Using a special form of Ulam's method, we estimate the measure-theoretic entropy of a triple (M,T,mu), where M is a smooth manifold, T is a C^{1+gamma} uniformly hyperbolic map, and mu is the unique physical measure of T. With a few additional calculations, we also obtain numerical estimates of (i) the physical measure mu, (ii) the Lyapunov exponents of T with respect to mu, (iii) the rate of decay of correlations for (T,mu) with respect to C^gamma test functions, and (iv) the rate of escape (for repellors). Four main situations are considered: T is everywhere expanding, T is everywhere hyperbolic (Anosov), T is hyperbolic on an attracting invariant set (Axiom A attractor), and T is hyperbolic on a non-attracting invariant set (Axiom A non-attractor/repellor).