The most common model of transistor operation is the current-amplifier model, like all models it is something of a simplification and it represents just one way of viewing the way a transistor works. I'll assume your question is in the context of a typical silicon NPN bipolar junction transistor (BJT) used in a common-emitter configuration.
For switching, I gather that the base needs a certain voltage to allow the current to travel through.
This is more or less correct, however the range of useful voltages is very small - you might as well think of it as fixed at 0.7 volts with tiny variations in normal use. This voltage is usually called Vbe (voltage across base and emitter). So long as the base voltage is enough to allow current into the base, the transistor can begin to operate. The more base current, the more current is able to cross from collector to emitter. like a valve, small amounts of base current allow large amounts of collector current. The ratio is known as the gain and might be 100x. Eventually the collector current reaches a maximum and the transistor is said to be in a saturated state. For switching applications you want to quickly drive the transistor into saturation (turning it "on") by quickly raising the base current to a suitable value.
For amplifying... Can only specific transistors amplify and specific ones switch?
Some transistors are better for some types of use. However many common small-signal transistors are used for both purposes.
are you using the power of your high current feed and in a sense mixing it in with you guitar signal to boost it?
No, that's not a useful way to think about what is happening. Think of a small flow of water turning a valve against a spring and controlling the a larger flow of water. It's about controlling not mixing.
if you had a pot to adjust the signal amplification, what exactly would you be adjusting, the current or voltage supplied.
Both usually. Voltage and current are dependent on one another. In the case of a resistive load they are related by the ohms-law equation V=IR. For a constant R, increasing V increases I (I=V/R).
Lastly I read that you can increase voltage and current with a transistor.
As I said, the most common model for understanding transistor behaviour describes the transistor operation in terms of current amplification. However you can usually find a way of converting a current into a voltage.
Is it possible to increase voltage and current above that of any of the supplies?
The voltage at the collector of a transistor in common emitter configuration cannot be made greater than the supply voltage (Vcc) by the normal operation of the transistor itself.
Note: With an additional inductor you can create a circuit that creates higher voltages but this isn't an intrinsic capability of the transistor itself.