Do electrons also gain mass by interacting with the EM field or all of its mass comes from the interaction with the Higgs field? I'm confused. I thought that the only reason why electrons had mass was that they interacted with the Higgs field, but I've been told that the EM field also contributes to its mass. However, if this is true, why don't photons also have mass (since it also interacts with the EM field)?
 A: ( This is rather a longer comment, so I posted it here as an answer, assuming it would help. )
EM field contributes to observed inertial mass of bodies which are composed of 2 or more charged parts, due to effect of self-force (one part exerts force on the other whenever the body accelerates, and these forces are delayed, which makes it behave as if it had more mass). This increase of mass is called electromagnetic mass.
It is not known how much of electron's mass is electromagnetic mass. In the early day of theory of electron (before special relativity was accepted), there was an hypothesis that whole electron mass was electromagnetic, but this didn't work out (electron would have to be much bigger than it is observed to be and there were other problems with the models of composite electron). Also, special relativity killed the initial motivation for "electromagnetic explanation" of mass, since any energy, not only electromagnetic, would imply increase in inertial mass. Thus if the electron components interact via forces different than EM forces (which they probably have to in order to maintain the size of the electron with microscopic limits), there would be increase/decrease of mass due to those forces as well.
The state of the knowledge is such that some part of observed mass of proton and neutron is electromagnetic and the same could be, in principle, true also for electrons. This contribution could be estimated from size of electron charge distribution in space, but so far in the experiments the electron behaves as if it is smaller than something like 10e-18 m, which is 1000x smaller than the classical electron radius.
If the electron is composite, this would imply that the EM mass of the electron is 1000x the observed mass of the electron or even more, so there would have to be another mechanism that decreases the mass to get to the observed value.
If the electron is point, the customary pragmatic view is that mass is an independent characteristic of the electron, which is not connected to its charge in any way.
AN ANSWER
Photons or EM waves do not interact with EM field; they are EM field. EM field can be thought of as having a mass if it is bound to a body and contributes to its inertial mass. For example, the bound electromagnetic field of charged ball means there is the electromagnetic mass effect described above and this influences the observed inertial mass.
But if the EM field is not bound to the body, like in case of EM waves (or photons) it is not considered to be part of the body and hence it has no influence on its mass.
A: This is a rather confusing point and I'm not sure if it has a good popsci-level explanation. 
In perturbation theory, it is a fact that if the electron is massless without the electromagnetic field, then turning on the electromagnetic field leaves it massless. This is because a massless electron has chiral symmetry, and this symmetry forbids a mass term from arising.
If you instead have a massive electron, where the mass is provided by the Higgs mechanism, the electron no longer has chiral symmetry and can pick up mass from the electromagnetic field. It picks up plenty of mass, in fact naively an infinite amount.
Similarly, in perturbation theory, the photon must remain exactly massless by gauge invariance. But if you started with a particle that was kind of superficially like a photon, but which had a finite mass, then indeed there's nothing stopping it from picking up a lot of mass, e.g. from the electron field and the electromagnetic field.
Why is there a discontinuity here? I'm not sure if there's an intuitive explanation. Here's a guess that might be totally invalid. The picture is that the electron carries around some excitation of the electromagnetic field with it as it moves, and this excitation must have positive energy. Then if we go to the rest frame of the electron, this energy contributes to the electron's mass by $E = mc^2$.
But now suppose the electron is massless. Then it must be moving at the speed of light, and it's impossible to go into its rest frame. You can try, by accelerating to try to catch up with it, but as you accelerate more and more, the energy in the electromagnetic field will redshift away. If you keep accelerating forever it will vanish entirely! So the contribution to the rest mass is zero.
A: Consider this as a long comment, too.
Electrons and photons are part of the standard model elementary particles, i.e. non composite point particles. They have a fixed invariant mass ( the length of the special relativity four vector), as seen in the table. The standard model encapsulates all the data acquired in particle physics experiments , and using it one can calculate and predict expected interactions , as the success of the discovery of the Higgs particle showed.

I thought that the only reason why electrons had mass was that they interacted with the Higgs field, 

Within the standard model, which is based on quantum field theory, the higgs mechanism gives mass to particles at the time of weak symmetry breaking. In this model, all the particles in the table are described with a field covering all space-time (x,y,z,t), and the vacuum expectation value,VEV, of these fields ** of these fields,except the Higgs field,** is zero. The Higgs field has a VEV of 246 GeV which is part of the mechanism gave mass to all particles  in the table, including the Higgs.
It is necessary to emphasize that the higgs mechanism is not an interaction. Symmetry breaking happened once as the energy of the universe fell due to the expansion at the weak symmetry break point, and the particles acquired a fixed mass at that point. As it is not an interaction it makes no sense to put the higgs mechanism and the electromagnetic interaction on the same footing as far as the fixed mass of particles is concerned.
Extensions and  different models may change these statements, but the main stream model is still the standard model.
A: From the standard model energy expression of the electromagnetic field and the estimated $r_e \leq 10^{-18}$ m for the electron radius, defined as the radius within which all charge is located, then the electromagnetic contribution to the mass should be at least $\Delta m_e = \frac{e^2}{4\pi\epsilon_0 r_e} \geq ~\sim 1.5 ~ GeV/c^2$. In order to arrive at $m_e = 511 ~ keV/c^2$ the Higgs contribution should be negative and nearly cancel the electromagnetic contribution.
