How should I connect the RX and TX pins of an Arduino to a Bluetooth module?
You need to use the  SoftwareSerial library that replicates the serial protocol on any of the PWM pins on the Arduino Uno.
I am guessing you are using the USB to power up Arduino.  This is not going to work with your Bluetooth module because it is connected to RX and TX pins on the Arduino. The RX and TX pins are used for USB communication and hence should only be used when you are not using the serial terminal on the Arduino IDE.