It might be worth while taking a step back.
Connect the BT module to GND and VCC, don't worry about connecting to the Arduino.
The LED should be blinking about 5 times a second.
Now remove the power, press and hold the little push button switch, reconnect the power (keeping the button switch closed). After seeing the LED come on you can release the button switch but don't release the switch too soon.
The LED should now flash on/off every 2 seconds or so; on for 2 seconds, off for 2 seconds. This means the module is in AT mode at 38400.
If you are still having issues post a clear photo of the board; front and back.
The garbage characters in the serial monitor can mean you have the wrong baud rate. I get similar results when I use software serial at 9600 and the board is set for 38400.
Have you have the module talking to another device such as an Android device?