How can higher current devices (motors, solenoids, lights, etc.) be controlled by an Arduino? I'm looking for a broadly applicable solution, one that can be adapted to a variety of projects.

I'm currently working on several projects that each require controlling devices ranging from 800mA to 2A from an Arduino Uno. One controls stepper motors, one controls 12vdc solenoid actuators, and one controls 12vdc pneumatic valves. 

For example:

The Arduino monitors a button, and each time the button is pressed it triggers the solenoid actuator. Because the Arduino is unable to source the current required by the solenoid, a seperate power supply is required with the Arduino controlling a switch (relay, transistor, etc.) that allows the higher current to pass. For the stepper motor, the layout is more complex as there would need to be four pins controlling four separate switches (to maintain interoperability of the circuit). The relay controls an air valve and requires 12vdc as well. 

I'm trying to figure out how to use a single circuit that can be used in each of these applications (and any future projects) that involve controlling higher current devices than the Arduino pins can handle. 

Prototyping speed, standardized components, and low cost are the driving factors. Switching speed, useful life, and noise are also important.

Is there a breakout board, circuit, or component that can be connected to an Arduino pin and used to control a high current device? Ideally with a software controlled potentiometer so that the resistance for different projects could be set in the sketch itself.
To drive such high currents, you may have to cascade several transistors (you can also use a Darlington transistor). There are arrays of Darlingtons mounted in a chip (e.g. the ULN2803A has 8 darlington transistors, but is limited to 500mA).

You probably will have to deal with higher power transistors; as an example I have found STMicroelectronics TIP110 which can support switching a 2A current (4A peak), but it would probably need a heatsink to dissipate heat.

Note that I wonder if your steppers really need 2A current (are they that big?). For steppers, you can generally find IC that can drive them easily, e.g. the L293D but this one can drive "only" 600mA).

As a conclusion, I am afraid you will not find a "one size fits all" solution, as all your devices are different and should be driven by the appropriate circuit.

Edit:

Since oversizing is not an issue in your prototyping case, then you could go with a MOSFET instead of usual bipolar transistors. A MOSFET will be able to drive higher currents and voltages than standard transistors. 

The downside is you can use it as a switch only (like e.g. a relay) and thus can't really drive the exact power for your devices. I guess that does not matter for a Stepper motor, or a Solenoid, but that may be important for driving lights for instance.

However, the good point is that you can still use PWM for that as the MOSFET switching speed is good enough for such purposes.

Regarding price, there are many diferent kinds of MOSFET out there, but I guess you can find one that fits yours needs (12V, 2A) for less than $1.

I advise you to take a look at this great article about this topic.