How does inertia work at the particle level? I've read recently that the mass of a proton is mostly not given by the Higgs mechanism. But rather it's given by the energy of quarks moving around inside the proton and gluons and other internal sources of energy. 
Obviously mass and energy are equivalent, so I'm assuming we can calculate the rest mass of a proton from the energy it holds. But wondering this made me question why exactly you get "inertia", or resistance to acceleration, at all from energy.
Why are things harder to move when they have more energy? Is there some process by which things with a lot of energy resist changes in motion? I feel like everyone just says $E = mc^2$ without actually asking why we experience the phenomenon of mass and inertia at all?
 A: A lot of the physical intuition for how inertia works in relativity comes from studying how mechanics works in SR. For example, the Wikipedia article here has a nice summary of some of the basics with the math using 4-vectors.
The physical intuition behind the idea of inertia comes down to the idea that if you apply a force to an object, there will be an acceleration of the object proportional to the force with 'mass' or 'inertia' being the coefficient determining how strong a change for a given force. Now in a non-relativistic world applying a force to a stationary object or a moving object will both yield the same change in velocity that you will observe. However, this is not true in relativity due to the fact that there is a maximum speed, $c$. So if you apply a force to a particle at rest, you will see a large change in velocity, but if the particle is already moving very close to $c$, it will appear to you to barely change at all. As an interesting side note, historically SR was taught with the idea of both a rest mass $m_0$ and a relativist mass $m=\gamma m_0 = m_0/\sqrt{1-(v/c)^2}$ (which is the mass in $E=mc^2$) to take the idea that the perceived mass becomes infinite as the speed of the particle approaches $c$ into account, but this has largely been dropped in modern physics.
A: 
Why are things harder to move when they have more energy? Is there some process by which things with a lot of energy resist changes in motion? 

This is not a general answer, but it should help give you a more intuitive feel for why this happens. 
Consider two identical boxes, both boxes have a perfectly reflective interior. One box is empty and the other contains a bunch of isotropic incoherent photons inside. Because of the photons the “full” box has more energy, and because photons carry momentum there is a pressure exerted on the walls. At rest this pressure is isotropic. 
Now, as you try to accelerate the full box the photons are blueshifted after colliding with the back wall and redshifted after colliding with the front wall due to the Doppler effect. This results in a net force on the rear wall which is larger than the net force on the front wall from the photons. This means that the box with the greater energy has more inertia. 
It turns out that, due to relativity, all forms of energy behave the same way.
