Six leds lights up sequence and back by pressing buttonup/down 5 times.

Other Hardware LEDs and Multiplexing
4

I used your setup and wrote only the main loop. This should be the full code:

const int buttonUp    = 9;
const int buttonDown  = 8;
const int ledPin1     = 7;
const int ledPin2     = 6;
const int ledPin3     = 5;
const int ledPin4     = 4;
const int ledPin5     = 3;
const int ledPin6     = 2;

boolean changeLed = false;
unsigned int ledCounter = 0; 

void setup() {
  pinMode (ledPin1, OUTPUT);
  pinMode (ledPin2, OUTPUT);
  pinMode (ledPin3, OUTPUT);
  pinMode (ledPin4, OUTPUT);
  pinMode (ledPin5, OUTPUT);
  pinMode (ledPin6, OUTPUT);
  pinMode(buttonUp, INPUT);
  pinMode(buttonDown,INPUT);
  Serial.begin(9600);  
}

void loop(){

if (button1Pressed ()==0 ){
  if (ledCounter < 5) {
    ledCounter++ ;
    changeLed = true;
  }
}

if (button2Pressed()==0){
  if (ledCounter > 1) {
    ledCounter-- ;
    changeLed = true;
  }
}

if (changeLed == true){
  digitalWrite(ledPin1, LOW); 
  digitalWrite(ledPin2, LOW); 
  digitalWrite(ledPin3, LOW); 
  digitalWrite(ledPin4, LOW); 
  digitalWrite(ledPin5, LOW); 
  digitalWrite(ledPin6, LOW); 
}

switch (ledCounter){
case 0 : digitalWrite(ledPin1, HIGH);break; 
case 1 : digitalWrite(ledPin2, HIGH);break; 
case 2 : digitalWrite(ledPin3, HIGH);break;
case 3 : digitalWrite(ledPin4, HIGH);break;
case 4 : digitalWrite(ledPin5, HIGH);break;
case 5 : digitalWrite(ledPin6, HIGH);break; 
default: digitalWrite(ledPin1, HIGH);break; 
}

changeLed = false;
delay(20);
}

int button1Pressed (void) {

  if (analogRead(buttonUp) == LOW){
   delay(3); // 
      if (analogRead(buttonUp) == HIGH){
       delay(3);    
         if (analogRead(buttonUp) == HIGH){ 
          delay(3);
             if (analogRead(buttonUp) == HIGH){ 
             return 0;
             }
         }
      }
  
  }
return 1;
}


int button2Pressed (void) {

  if (analogRead(buttonDown) == LOW){
   delay(3); // 
      if (analogRead(buttonDown) == HIGH){
       delay(3);    
         if (analogRead(buttonDown) == HIGH){ 
          delay(3);
             if (analogRead(buttonDown) == HIGH){ 
             return 0;
             }
         }
      } 
  }
return 1;
}

It's not tested. The debouncer (the multiple reads to see if he pin is in high; error protection) might be a little bit to "harsh". If the pressed of a pushbutton is not always seen you can start looking there (in the buttonxPressed() functions).

Good luck