There are a number of quantities that physics has found to be conserved.
"Conserved" means that if you take a situation, and you measure what is in it, then something happens in it (where you measure all inputs and outputs), the measure of those quantities doesn't change.
- Linear Momentum
- Angular Momentum
- Center of Momentum Velocity
- Electric Charge
- Color Charge
- Weak Isospin
No experiment has proven that any one of the above is not conserved in any examined situation.
Under Noether's Theorem (proven by Emmy Noether, one of the most important mathematicians in history in my opinion), each of these conservation laws lines up with a symmetry in the mathematics of physics. Mass-Energy conservation, for example, can be shown to be mathematically equivalent to suitably constructed laws of physics not caring what time it is when it makes predictions.
The kind of "symmetry" Noether is talking about is a generalization of the symmetry you are used to. Some of the symmetries are exotic compared to what you might be used to.
The thing about Mass-Energy conservation is it means that it sort of implies that the two (mass and energy) are somehow the same thing. You can convert one to the other (with effort) and back again.
If you actually start looking really deeply into "solid matter", it ain't very solid. Much of what you consider "solid" is due to Pauli exclusion principle and the lowest energy states for Fermions crowding alternatives out, and increasingly high amounts of pressure being required to shove more Fermions into the matter.
Going deeper, electrons don't have much mass-energy; most of the mass-energy of "solid" matter comes from Neutrons and Protons. They, in turn, are made out of Quarks; most of their mass-energy isn't from the Quarks they are made out of, but rather the potential energy in the binding of the Quarks to each other. (There remains a very small "rest" mass that isn't produced by such binding)
It turns out that if you make a perfectly insulated box, and you heat it up, the box gets heavier. If you build a really powerful spring inside of it and you squeeze it shut, the box gets heavier. The degree that this adds weight and mass is tiny in most practical situations.
Atomic nuclei are basically tiny nearly weightless stuff that has a ridiculously powerful spring tightly coiled in it, and almost all of the mass of "stuff" comes from the tension on the spring, not the things that the springs are attached to.
And this doesn't mean that the things the spring is attached to -- the quarks -- are somehow not also energy. If you take a bunch of quarks and slam them into each other really really hard, you get a LOT more quarks appearing. They spew off in every direction.
The conservation laws above all hold in this collision, but we don't conserve "number of quarks". We conserve difference of quarks and antiquarks, we conserve color numbers, we conserve mass-energy, etc.
In that way, everything is energy. Also, in a similar way, everything is angular momentum. But the second statement is sort of less believable than the first.
Now, most conserved things are signed values or vectors. Mass-energy tends to be denoted as a positive value (but see stuff like the Casmir effect; the zero point of mass-energy is nearly arbitrary). Conservation laws care about the change in Energy in the system, not the total.
Strange things are predicted to occur at the limits of this. If you take a complete vacuum, you'll see one thing. If you start accelerating fast enough in that vacuum, the math predicts that you'll experience a "heat bath" of energy being emitted from the vacuum. Such a "heat bath" includes particles that you won't experience if you where not accelerating.
So, it is plausible that "everything is energy" doesn't go far enough.