cycling 4 LEDs at varrying speeds

Using Arduino Project Guidance
55

The code above seems to do what you asked for. This code:

int timer = 700;           // The higher the number, the slower the timing.

void setup() {
  // use a for loop to initialize each pin as an output:
  for (int thisPin = 2; thisPin < 6; thisPin++)  {
    pinMode(thisPin, OUTPUT);      
  }
}

void loop() {
  // loop from the lowest pin to the highest:
  for (int thisPin = 2; thisPin < 6; thisPin++) {
    // turn the pin on:
    digitalWrite(thisPin, HIGH);  
    delay(timer);                  
    // lower the delay if timer >= 20       
    if (timer >= 20) {
      timer = timer - 1;
    }

    // turn the pin off:
    digitalWrite(thisPin, LOW);    
  }
}

I tried it out, the LEDs flash in sequence, gradually (very gradually) getting faster.

Is that not what you wanted?