Each press of the button cycles the LED colors. It turns off after red.

Just wire it up and upload the following code.

int redPin = 3;
int greenPin = 5;
int bluePin = 6;
int buttonPin = 2;
int initialCount = 0;

//uncomment this line if using a Common Anode LED
#define COMMON_ANODE

void setup()
{
  pinMode(redPin, OUTPUT);
  pinMode(greenPin, OUTPUT);
  pinMode(bluePin, OUTPUT);
  pinMode(buttonPin, INPUT_PULLUP);
   setColor(0,0,0);
}

void loop()
{
  int buttonState = digitalRead(buttonPin);
  if ( buttonState == LOW ){
    delay(500); //delay for debounce
    initialCount++;
    if (initialCount > 4 ){
      initialCount = 0;
    }
    switch (initialCount){
      case 0:
        setColor(0,0,0);
        break;
      case 1:
       setColor(0, 255, 0);  // green
        break;
      case 2:
        setColor(0, 255, 255);  // aqua
        break;
      case 3:
       setColor(0, 0, 255);  // blue
        break;
      case 4:
       setColor(255, 0, 0);  // red
       break;
    }

  }

}

void setColor(int red, int green, int blue)
{
  #ifdef COMMON_ANODE
    red = 255 - red;
    green = 255 - green;
    blue = 255 - blue;
  #endif
  analogWrite(redPin, red);
  analogWrite(greenPin, green);
  analogWrite(bluePin, blue);  
}