Communicate with bluetooth module from Arduino Uno

I'm trying to get my bluetooth module into command mode. I have pin 0(RX) on the Uno connected to the UART_TX pin on the bluetooth module, and pin 1(TX) on the Uno connected to the UART_RX pin on the bluetooth module.

Heres the sketch thats running on the Uno.

int incomingByte = 0;   // for incoming serial data

void setup() {
  // put your setup code here, to run once:
  Serial.begin(9600);   // opens serial port, sets data rate to 9600 bps
  Serial.flush();
  //enter command mode
  Serial.println("$$$");
}

void loop() {
  // send data only when you receive data:
  if (Serial.available() > 0) {
    //read the incoming byte:
    incomingByte = Serial.read();

    //say what you got:
    Serial.print("I received: ");
    Serial.println(incomingByte, DEC);
  }
}

This is what I see on the Serial Monitor

$$$

I expected to see

$$$
I received: C
I received: M
I received: D

What am I doing wrong? Or if its not supposed to do what I expected, what am I not understanding correctly?