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?
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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.
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