I'm not sure I can answer that without confusing things more, but I'll see what I can do. In an MOS transistor, the gate is insulated, so there is no base current, just voltage. This voltage controls the transistor.
In a bipolar (NPN or PNP) transistor, the current through the base causes a larger current through the collector, amplifying by the beta factor. So the transistor is amplifying current. But the current depends on the voltage between the base and emitter, so from that perspective the voltage controls the transistor too.
Whether you're amplifying current or voltage depends on the circuit, so I can't give more than a handwaving answer.
In field effect transistor (in which the actual physics involved are at least for me simpler to grasp) the gate is isolated from everything else and voltage at the base directly changes the geometry of the conductive channel between the S and D pins. In effect the gate voltage directly influences the resistance of the component. MOS is an name for particular practical realization of this mental model.
In bipolar junction transistor (ie. PNP/NPN) there are two diodes that are positioned just so that conduction of one of them influences the other in such a way that when one is positively biased the other will conduct even when reverse biased. For the typical BJT these two diodes have significantly different construction and thus there is difference between emittor and collector, but the effect works both ways (and in fact many circuits will somewhat work even with the 2N3904 connected the wrong way around). The effect is also caused by change of properties of doped semiconductor material in response to electric field gradient but (at least for me) there is no directly applicable model involving discrete lumped components changing their parameters in response to external stimuli that matches the underlying physical principle.
> I understand that transistors amplify current but how do they amplify voltage
Any voltage through a resistor will produce a current, and any current through a resistor will produce a voltage.
By properly connecting resistors at the transistor base and collector, one can turn a driving voltage into a current and the collector current back into an output voltage. The basic common emitter transistor amplifier circuit is a good example as it shows how a current amplifier like a transistor is used to amplify a voltage. Resistors are the secret that allow all permutations: voltage to voltage, current to voltage, voltage to current, current to current.
Transistors change their effective resistance based on the "control stimulus", which in turn changes how much current can pass through them.
In BJT transistors the "control stimulus" is the current flowing through the base pin, while for (MOS)FET transistors it's a voltage potential between the gate and source pins.
The amplification happens because in the right region the change in effective resistance is high for small variations in the "control stimulus".
If you drive a BJT with a resistor in front of the base pin, you can drive it with a voltage. If you put a resistor between the gate and source pins, you can drive a MOSFET with a current source.
So the way you control them is different, but what they end up doing is the same. And by using a resistor you can effectively change the way you drive them.
Your DC battery or power supply provides your headroom, and the transistor Base or Gate senses a small increment of that voltage and can sometimes deliver as much current as it takes to push the voltage across a resistance or impedance right up to the rails.
In a bipolar (NPN or PNP) transistor, the current through the base causes a larger current through the collector, amplifying by the beta factor. So the transistor is amplifying current. But the current depends on the voltage between the base and emitter, so from that perspective the voltage controls the transistor too.
Whether you're amplifying current or voltage depends on the circuit, so I can't give more than a handwaving answer.