Draw to the screen?

Setup

There's quite a bit going on in the setup, but it thankfully is mostly boilerplate.

The functions you want, in order, are:

VIDEO_Init(): This needs to be called before any other VIDEO functions. Documentation says it should be done "in the early stages of your main(), but we've had success in moving it to a different function that is called from main.
VIDEO_GetPreferredMode(): This returns you your GXRModeObj struct pointer (a "rendermode object") defining the screen layout and mode. What you get from this depends on the console settings, and if it's PAL or NTSC, things like that. The parameter also takes a GRXModeObj and seems to overwrite it if present? Not sure why you'd want that, just pass NULL.
SYS_AllocateFramebuffer(): Allocate cacheline aligned memory for the external framebuffer based on the passed rendermode object. Returns a 32-byte-aligned pointer to the framebuffer's start address. This allocates the memory for our framebuffer, but we cannot use it yet.

MEM_K0_TO_K1(): This is a macro to cast a cached virtual address to an uncached virtual address. In this case, we use this to convert the address of the allocated framebuffer to one we can use to write to it.

VIDEO_Configure(): Configures the "VI" (video interface?) with the given rendermode object. Internally, this is setting a lot of video registers with our rendermode settings

VIDEO_SetNextFramebuffer(): This sets some video register to point to our allocated framebuffer memory. In effect, we are telling the video hardware where the framebuffer is.

VIDEO_SetBlack(): TRUE to black out the screen, FALSE not to.

VIDEO_Flush(): When we make changes to the video hardware registers in the above functions, we do not actually make those changes to the hardware until we flush them with this function. This in effect commits our changes.

VIDEO_WaitVSync(): Waits for the next vertical retrace.

Putting all that together, a simple (minimal) video setup would look like:

#include <gccore.h>

static void* xfb = nullptr;
static GXRModeObj* rmode = nullptr;

void video_initialise()
{
	// Initialise the video system
	VIDEO_Init();

	// Obtain the preferred video mode from the system
	// This will correspond to the settings in the Wii menu
	rmode = VIDEO_GetPreferredMode(NULL);

	// Allocate memory for the display in the uncached region
	xfb = MEM_K0_TO_K1(SYS_AllocateFramebuffer(rmode));

	// Set up the video registers with the chosen mode
	VIDEO_Configure(rmode);

	// Tell the video hardware where our display memory is
	VIDEO_SetNextFramebuffer(xfb);

	// Make the display visible
	VIDEO_SetBlack(FALSE);

	// Flush the video register changes to the hardware
	VIDEO_Flush();

	// Wait for Video setup to complete
	VIDEO_WaitVSync();
	if (rmode->viTVMode & VI_NON_INTERLACE)
		VIDEO_WaitVSync();
}

Draw to the screen?

Setup

Text output