>This made me visualize photons(as an example of quantum) as energy carrying particles, which can be absorbed by atoms or molecules one at a time.

In the [Standard Model of particle physics][1] photons are zero mass particles on an equal axiomatic level as electrons, for example.In mainstream physics the SM is the basic frame of particles and from these all other physics theories  emerge (except  gravity and general relativity,it is still not included in the quantum theories with strict mathematics) 

>But, does this mean there is a delay in time between absorption of one photon and the next one? I think there is as my book clearly says discrete chunks of energy. Also, if the answer is yes, how long is it, or what does it depend on?

Matter is composed out of zillions of particles. [Avogadro's number][2] gives $6.02214076×10^{23}$ elementary entities per mole of matter.

There are several ways photons can be absorbed in matter, 1) individual photons interacting with atoms and changing the energy level of the electrons 2)individual atoms interacting with molecules and changing the quantum configuration energy levels3) with a lattice consisting of zillion of atoms again changing the energy levels.4) They can scatter with the above and lose energy. (The end result is that the impinging photons disappear into raising the heat of the sample under study but this does not involve the question )

Individual photons are absorbed by individual atoms/molecules/lattice  in the particular mass one is studying.

>But again, we have learnt that light is a wave, and a wave is continuous(If I am not wrong), so when considering quantum physics, do we ditch the electromagnetic wave properties of light or something?

In physics we do not ditch any theory that is working in its own area of validity. Maxwell's equations describe light, electromagnetic waves, very well for dimensions where quantum effects are very small. The build up of classical waves by photons is a complicated mathematically form that one can understand when studying physics further, as it needs [quantum field theory.][3] 




  [1]: https://en.wikipedia.org/wiki/Standard_Model
  [2]: https://en.wikipedia.org/wiki/Avogadro_constant
  [3]: https://en.wikipedia.org/wiki/Quantum_field_theory