On many sculpting tutorial videos when they enter sculpt mode their mesh becomes a rich red as in here. On mine, it remains in the same black/white layout as in edit mode here. It's much easier for me to see gradients on the red so how do I go about changing this?
You can change the Viewport Shading in the top right corner of the 3D editor. Open up the drop-down menu by pressing the upside-down ^, then select MatCap. Then, press the ball to change which MatCap you want to use.
I believe this is the one you're seeing in the video.
Note that these settings will only be visible if you're in the Solid Viewport Shading mode.