In setup(), you'll want to initialize Bridge and start a serial connection. Before running the rest of setup() wait for a serial connection to become active.
The key is that you need to call Bridge.begin() before doing anything with a Process class object (or any of the other classes that are in the Bridge Library.
Opening up the serial port is not a required part of the using the Process class. That example has you open the serial port because it is using the Process class to call the
curl command to retrieve a text file from an HTTP server. It then displays the retrieved data on the serial port. For your use, you do not need the serial port, unless you want to use it for debugging purposes or to see any error messages returned by the
curl request.
So when you add the Bridge.begin(); and Serial.begin(9600); under void setup(), does that mean that a serial monitor window will randomly pop up or is there something else I'll need to do to get a serial monitor session working?
It will not automatically open a serial monitor window. You will need to manually open the Arduino Serial Monitor from within the Arduino IDE, or you will have to use some other serial terminal emulator program. That is, of course, if you decide you want to use the serial output.
Because this example sketch immediately loads the remote text file, and only does it once, it includes this line of code:
while (!Serial);
This causes the sketch to wait until a USB serial connection is established - in other words, for the Serial Monitor to be opened. This way, the output from the Process command will be visible. Without this delay until a connection is made, the output would likely be displayed before you had a chance to open the serial monitor, and you would never see the output.
While that bit of code can be handy for debugging, you will probably not want it in your final sketch - if you include it, you would need to open a serial monitor each time you power up or restart your sketch. That's generally not a normal operating scenario for a finished product.
Also- is this the standard m.o. for serial monitoring with the Dragino Yun?
No, that's the standard debug output method for the official Arduino Yun board. That board uses an ATmega32U processor which has a USB serial port (Serial) for use by the Serial Monitor over the USB connection, and a second hardware serial port on pins 0 and 1 (Serial1) that talks to the Linux processor using the Bridge library.
The Dragino Yun Shield is different - it doesn't have an AVR processor to run a sketch, instead it is a shield to connect to an Arduino processor board. Since there are different boards it can be used with, and they have different processors, the ability to use a serial port for debugging and the Serial Monitor will vary. For example, if you are connecting the shield to an Uno, the only serial port available is connected to pins 0 and 1, which will need to be used for Linux communications with the Yun Shield. That means that the USB serial port needs to be disabled and Serial Monitor communications are not possible. If you are using it with an Arduino Leonardo, then that will be much closer to the official Yun, and you will have the USB serial port that you can use with the Serial Monitor. Other Arduino boards will have different limitations and abilities when used with the Dragino Yun Shield.
The Dragino site will be the prime place to go for information specific to the Dragino Yun Shield. That shield is similar to an official Yun, especially when connected to a Leonardo, but there are differences. Any Bridge library examples that come with the Arduino IDE or from this site will assume that you are using an official Arduino Yun, so some changes and allowances will have to be made to accommodate the differences from the Dragino Yun Shield. I have no direct experience with that shield, so I can't give you specific changes that will need to be made.