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void setup()
 size(800,800);
 background(0);


void draw() 
 intx= 20,40,60,80,100,120;
 fill (255,0,00);
 for (int i = 0; i <x.length; i ++)
 ellipse (i*6, 100,10,10);
 

I am trying to get the "worm" to move horizontally in a straight line across my canvas. Attached is the "worm." I don't know where to start with the movement because I need to move the whole array at once. Thank you!

Store the current positions of the worm in a global array, where the head is the first position.

int x= 30,24,18,12,6,0;
int step = 6;

Change the position in every frame by the distance between 2 parts of the worm. Shift the current positions in the array in a loop:

draw() 

 .....
 for (int i = x.length-1; i > 0; --i )
 x[i] = x[i-1];
 
 x[0] += step;
 .....

Chang the direction if the end of the window or the begin of the window is reached, by the head of the worm:

if ( x[0] >= width || x[0] < 0)
 step *= -1;

Draw the worm in the reverse order:

for (int i = x.length-1; i >= 0; --i )
 ellipse(x[i], 100, 10, 10);

The full code may look like this:

void setup()
 size(200,200);


int x= 30,24,18,12,6,0;
int step = 6;

void draw() x[0] < 0)
 step *= -1;

See the preview:

Moving an array of values isn't much different from moving a single value.

Let's look at how we'd move a single value:

float x = 0;

void draw()
 background(0);
 ellipse(x, height/2, 20, 20);

 x++;

To move an array of values, you'd do the same thing, but for each index of the array. Something like this:

float xArray = 20, 40, 60;

void draw()
 background(0);

 for(int i = 0; i < xArray.length; i++)
 ellipse(xArray[i], height/2, 20, 20);
 xArray[i]++;
 

Shameless self-promotion: here is a tutorial on animation in Processing, and here is a tutorial on arrays in Processing.