I know that it is believed that energy is discrete, in that it travels in quanta. I was wondering if there is any evidence which either proves or disproves something similar with both time and distance?
As far as we can tell (up to energy scales we've measured so far), spacetime is a nice and smooth manifold. It might happen that the smoothness is approximate and spacetime is discrete at a much more microscopic scale, or it could turn out that spacetime is smooth all the way through. Short answer: We don't know.
About the notion of energy quantization: Energy can only be transferred in quanta, but the size of the quanta are not fixed. You can "tune" the energy of a photon by tuning it's momentum/frequency. Once you fix the momentum, then energy can be quantized only in multiples of that quantum. It's a bit like buying (say) rice... you can buy it only in packets, but you can put how much ever you want into one packet.