Say I buy a device that requires 25 volts to function, but I want to control it using my 5v arduino. How would I go about powering/controlling higher voltage devices?
Yes it's possible, and it's usually done by step-up (boost) converters. These converters are based on the principle that inductors react to sudden changes in their current with a high voltage. So, an inductor is put in series with the supply and switched (here the name Switching Mode Power Supply, SMPS) between a short circuit and the load. When the switch opens, the inductor has to discharge the stored energy and it does that creating an opposite voltage than the one it's charged with.
Another efficient way is also the charge pump, where capacitors are charged at the supply voltage (5V in this case) and then put in series with the supply rail to generate the higher voltage. It's commonly used to double voltage, but different ratios are possible.
All these circuits create a pulsed output voltage; it can be filtered afterwards to give it a more stable shape if required.
An example of the answer by Petteri Hietavirta can be seen at the link.
Opto-Isolator is a safe way to go to avoid frying the Arduino.
If you have correct voltage supply available for the device you want to control with Arduino, check optoisolators.
With that your Arduino should be safe from high voltage but still able to control the device.