Arduino, built-in led&TX issue related (i think with) => Serial Port (Serial.beg

Using Arduino IDE 1.x
1

hiii there!!! = )
I'm new on this, and feeling useless lol
I'm following button example... ^^'
first of all, i do the circuit, and upload the example, and the built in LED and number 13, is on constantly, but when I press the button it brights more.... (the code is the same as button example...)
seems like it's entering the first condicion of "if (buttonState == HIGH){ (?!?!??!?!?)

example code:

// constants won't change. They're used here to set pin numbers:
const int buttonPin = 2;     // the number of the pushbutton pin
const int ledPin =  13;      // the number of the LED pin

// variables will change:
int buttonState = 0;         // variable for reading the pushbutton status

void setup() {
  // initialize the LED pin as an output:
  pinMode(ledPin, OUTPUT);
  // initialize the pushbutton pin as an input:
  pinMode(buttonPin, INPUT);
}

void loop() {
  // read the state of the pushbutton value:
  buttonState = digitalRead(buttonPin);

  // check if the pushbutton is pressed. If it is, the buttonState is HIGH:
  if (buttonState == HIGH) {
    // turn LED on:
    digitalWrite(ledPin, HIGH);
  } else {
    // turn LED off:
    digitalWrite(ledPin, LOW);
  }
}

circuit picture :

986×688 128 KB

then i've added a printing(button on /off) to see whats going on...to see when I press the button or not...
as soon as I put Serial.begin(9600); it starts blinking the TX and build-in led....

801×416 33.7 KB 

// constants won't change. They're used here to set pin numbers:
const int buttonPin = 2;     // the number of the pushbutton pin
const int ledPin =  13;      // the number of the LED pin

// variables will change:
int buttonState = 0;  
int PbuttonState = 0;
// variable for reading the pushbutton status

void setup() {
  
  Serial.begin(9600);
  
  // initialize the LED pin as an output:
  pinMode(ledPin, OUTPUT);
  // initialize the pushbutton pin as an input:
  pinMode(buttonPin, INPUT);
  
}

void loop() {
  // read the state of the pushbutton value:
  buttonState = digitalRead(buttonPin);
  
  // check if the pushbutton is pressed. If it is, the buttonState is HIGH:
  if( PbuttonState != buttonState ) {
  if (buttonState == HIGH) {
    // turn LED on:
    digitalWrite(ledPin, HIGH);
    Serial.println(" BUTTON ON ");
  } else {
    // turn LED off:
    digitalWrite(ledPin, LOW);
    Serial.println(" BUTTON OFF ");
  }
  }
  PbuttonState = buttonState; 
}

i've checked everything,
-changed the button
-changed of arduino,
-changed cable
-added resistors to check maybe it could be ground issues... :(((

really don't know what else to do :(((((((( really bad start

thanks for your help and time....