How should I think about voltage and power?

First of all potential difference in free space in an electric field. I prefer to define pd between two points, P and Q, as the work done by the electric field on a charge, per unit charge, as it goes from P to Q. So a greater pd between the points implies more work done on the charge as it goes from P to Q, which can only mean a greater electric field strength, that is a greater force acting on the charge. [Work = Force x distance in direction of force, and we're considering the fixed distance between P and Q.]

So if you apply a pd between the ends of a wire, the free electrons in the wire experience forces, urging them to travel through the wire. Because of collisions between the electrons and the lattice of ions (this is simplified) the electrons don't accelerate continuously under the force from the electric field, but reach a steady mean speed (called the drift speed). If you increase the pd you increase the force on each electron and the drift speed increases. This means that more electrons pass through any cross-section of the wire per second, that is the current increases.

Jumping now to the end of your question: "Why does POTENTIAL energy per charge between two points translate to THERMAL energy? I thought that if voltage was a push, the potential energy would just get converted to kinetic energy in the electrons, so where did the thermal energy come from?"

(1) Voltage isn't "a push"; its units are joules per coulomb! But, as I tried to explain above, it is related to the push (that is the force) that charges get in an electric field.

(2) The thermal energy comes from the collisions that the electrons, driven by the electric field and losing electrical potential energy, make with the lattice of ions. This increases the random vibration energy of the ions. [The extra kinetic energy that the electrons acquire due to the voltage applied is pretty negligible. For a current of a few ampère in an ordinary wire, the drift speed is in the order of a millimetre per second.]