The R and T in "Rx" and "Tx" stand for "Receiver" and "Transmitter".
So the Transmitter from one device should be connected to the receiver of the other device. Thus:
- Arduino Uno Rx -> HC-06 Tx
- Arduino Uno Tx -> HC-06 Rx
Thanks to the pictures that I can now see, what are the values of the resistors on the Arduino Tx pin? 2.2k I think, but I cannot see very well.
In idle mode, both Tx and Rx should be high (5V). Can you verify this with a multimeter? This pdf (p.8) states that the Rx should be pulled high. I assume that is what you did with the resistors, but I am confused to why you have two resistors. Could you explain this please? Usually a simple 10k resistor to Vcc should be sufficient. Adding a resistor between the Arduino Tx and the cable will cause reflections because of the impedance change and affect the signal quality.