I learnt that only electrons take part in chemical reactions and the protons do nothing (neutrons are electrically neutral, so they do nothing). Why is this so if the protons are positively charged and without them, electrons in the other atom won’t be attracted (and since electrons are like charges I do not think they would interact with each other)?
The nature of an electron is more complex than a single charged particle. Once you get down to the quantum level, our intuition about how matter interacts with other matter fails.
A simple answer to your question is that electrons occupy a much larger volume than nuclei. When two atoms get in close proximity to one another, their electrons will "bump into one another" and interact far before their nuclei will.
You are also correct that the nucleus of a neighboring atom is essential for the electrons to interact with one another. If there was no positive charge to attract another electron, a bond would not be formed. The point your instructor was trying to get across is that in most (excluding nuclear reactions) chemical reactions, electrons are the species moving around and forming/breaking bonds.
If you would like a more in-depth discussion on the true quantum nature of electrons and their interactions, you should check out Mulliken's Nobel Lecture from 1966. It is a great introduction.
Because chemistry is all about bonds and bonds are made from electrons
A reasonable definition of chemistry is the science of how substances change and react. Most of those changes and interactions are caused by changes in bonds (rearrangements,breaking and forming new bonds). So, since most chemistry is about what happens to bonds, most chemistry is about electrons.
But the number of electrons and the way they organise themselves in space is determined by the nuclear charges in the atoms involved in the bond(s). It isn't that protons are not relevant to the bonds, but they don't get close or involved when the electrons are rearranging themselves in chemical reactions.
And electrons do interact with each other not least because of the Pauli interactions that prevent more than two electrons occupying an orbital (orbitals are a way of describing the cloudy electron distribution in space and bonds are, to oversimplify a little, made from orbitals that span more then one atom).
In an atom, electrons also occupy orbitals and fill them up as more are added to a "naked" atom in order of the energy levels. Higher energy orbitals tend to be further from the nucleus than the lowest energy ones but electron interactions of a different sort matter here too as the "lower" orbitals shield the higher ones from some of the electric charge in the nucleus. Molecules are a bit more complicated, but the shielding still matters. Most chemistry happens with the outer orbitals/electrons and the inner ones in one sense "protect" the nucleus from any of the chemistry that is happening as electrons skip about to break or create bonds.
Also, in spatial terms, most of the "action" is happening a long way away from the nucleii, so they mind their own business and don't get involved.