If a voltage differential causes electrons to move through a conducting wire, and no other forces are acting against them, will they always move at the highest possible speed through that material?
Is it binary in the sense that they're either not moving, or moving as fast as possible?
If it's not binary, is it possible to apply a low enough voltage to move the electrons at half, or a quarter of their maximum speed?
We should distinguish electrons' usual random motions from their drift motion.
With no potential difference and thus no net "push" in any direction, the electrons are not stationary. If you look at one electron, it rapidly at very high speed (hundreds of $\mathrm{km/s}$) flies around and bumps into neighbors, causing it to change direction constantly and fly around randomly. This motion is due to thermally induced kinetic energy. Because every electron behaves in this way and since they all move randomly in every direction, the net motion averages out to zero. The current, which describes the net motion, is thus zero, $$I=0$$
With a potential difference, and thus a net "push" on all electrons in the same direction, this random motion still takes place. But at the same time they all feel a push in the same direction giving them all the same acceleration in the same direction on top of the random motion. The result is that while they randomly fly around they at the same time shift - or drift - slowly according to this "push". The net drift speed $v_d$ is small (fractions of $\mathrm{mm/s}$). And this drift speed is what describes current: $$I=neAv_d$$ $n$ is electron concentration (how many per volume), $A$ cross-section area of the wire and $e$ the electron's charge.
Now, the larger the "push", the faster the drift speed, and thus the higher the current. This is not a binary value. $v_d$ is reached after the electrons' net motion has accelerated up and stabilized due to the resistance of the circuit. In other words, they gradually reach this stable drift speed (steady current), which can be any value depending on the circuit resistance $R$, the wire material and the "push" (the potential difference $V$), which Ohm's law tells us:
$$I=\frac VR\quad\Leftrightarrow \quad v_d=\frac V{neAR}$$
The electron net motion $v_d$ is not binary.
If a voltage differential causes electrons to move through a conducting wire, and no other forces are acting against them, will they always move at the highest possible speed through that material?
There is no such "highest possible speed" (except for the ultimate maximum speed speed of light). If a voltage is "pushing" electrons up in speed, and there is no resistance, then they will accelerate forever. For example, a low-resistance wire connected to a battery will soon melt due to the very high current that continues accelerating.
