How do electrons move from one energy level to another?
There are misconceptions in your learning experience, (you do not give your physics background).
So having learned that electrons can move from one energy level to another by transmitting or taking in energy,
It is not the electrons that are moving in the classical sense. It is the whole atom which has quantum mechanical solutions with energy levels and orbitals for the electron. The atom absorbs a photon of an energy between two energy levels and the electron goes to the higher energy orbital. Orbitals are not orbits. They are probability loci, of where in (x,y,z) the given electron would be if measured. See this calculation of the orbitals available to the electron of the hydrogen atom.
As for the time taken, the energy levels have a width, and that corresponds to a lifetime, so there is nothing instantaneous.
the electron is destroyed in the original orbital, then recreated in another orbital, and the result also releases a photon. I thought that matter could not be created nor destroyed.
This is a confused use of quantum field theory, which is not very usefull for atomic spectra calculations. QFT describes well high energy scatterings, not the bound states of atoms and molecules.
There is nothing sacred about matter at the level of quantum mechanics and special relativity. Mass is not a conserved quantity, it is the four vector algebra that describes the creation and annihilation of particles . If there is enough energy in an interaction to create a particle antiparticle pair , quantum mechanics gives a probability for this to happen.
As for the last paragraph, Quantum field theory is not very useful for bound states as atoms and molecules, as its calculations involve a series expansion. Atoms can have complete solutions so to use the model of creation and annihilation operators on the corresponding fields makes no sense. There are such calculations but as the answer here says, still in a research stage,
You may be horrified or delighted to know that this question at least partially falls into the domain of interpretation.
Electrons are not particles that move in classical trajectories (unless you are a devotee of Bohm). They generally don't even have a well defined position, so they don't "jump" in any physical sense.
Experimentally the energy levels are discrete (which match the QM calculations) so the electron's energy does undergo a "jump".
An atom may find itself in a situation where the probability of electron emitting or absorbing a photon steadily increases, in which case the waveform steadily changes shape, but measurement shows that the energy levels are always at the before or after levels and never in between. In that sense, the energy jump is "instantaneous".
Many eminent physicists make comments like "particles are excitations of the quantum field" because that helps them think about the physics, but that is somewhat glib. The reality is more complex. Similarly many believe that the waveform always changes smoothly but that is really inconsistent with phenomena like double slit diffraction. These debates have no winners so it is much easier to follow Feynman and just "shut up and calculate".