I want to ask how electrons travel though the different orbitals?
Electrons are quantum mechanical entities,they cannot be described by the motions of classical billiard balls. That is why the terminology "orbital" was invented. We only know about the electron around the proton in the hydrogen atom that when probed, it has a probability of being within the orbital solution.
As far as the mathematics goes there is only the probability and no continuity between two space points so that a classical velocity could be defined.
They can only have discrete amount of energies (energy levels). If it doesn't have enough energy to go to higher energy level,
The electron remains at its energy level, unless a photon with the appropriate energy hits the atom. Orbitals might mathematically overlap in space, particularly if there are many electrons, this does not mean there is an energy exchange, it just means that the probabilities of two energy orbital can overlap in space. Only if there is a probability for an energy transfer to a lower state then the space overlap has a meaning. This is how electron capture happens with nuclei , the S level overlaps with the nucleus and there exists a probability for the electron to be captured by a proton given the energy balances of the system.
it will come back to the lower one. This happens instantaneously.
See above. The electron is not moving like a billiard ball. Its path is not consecutive. There are only probabilities.
There is nothing instantaneous in quantum mechanics . If one wants to go to the details of "electron interacting with a photon" one has to go to quantum field theory, where the interactions are described with Feynman diagrams and there are constants characterizing them which define the order of magnitude of the time transition.
So does it move with infinite velocity within two energy levels during transition?
No , there are no infinities . If the electron is in an energy level that can release a photon and relax to the lower level, there exist time constants coming from the probability distributions which give a lifetime for the decay and a width to the energy line.
If it is at the ground state it will stay in the ground state forever, unless an appropriate energy photon hits the atom, either to kick it up an energy level or free it from the material completely, as in the photoelectric effect.