We study chaotic dynamics of nonlinear oscillators with the form of a two-frequency quasiperiodic perturbation of a planar Hamiltonian system possessing a homoclinic orbit whose interior contains a one-parameter family of periodic orbits. In the extended phase space the unperturbed system has a three-dimensional homoclinic manifold and a one-parameter family of invariant 3-tori. Using Melnikov's technique and the second-order averaging method, we show that chaotic motions may exist near the unperturbed homoclinic manifold and the unperturbed resonant tori. These chaotic motions result from transverse intersection between the stable and unstable manifolds of normally hyperbolic invariant tori, and are characterized by a generalization of the Bernoulli shift. We also give an example for the quasiperiodically forced Duffing oscillator and demonstrate the existence of these chaotic motions by numerical simulation.