Can the Uno handle two serial port functions ?

Thank you Pylon-

  pinMode (0, INPUT);  // rxd
  pinMode (1, OUTPUT);  // txd

You must not do that. Once you called Serial.begin() pins 0 and 1 are handled by the hardware you must not call any functions that change the hardware configuration.

I have tried this code as shown above and with the pinmode lines commented out
and have not seen a difference in the operation or behavior of either one.

Or do I need to add some sort of polling code so that data gets echoed back to the serial device also?

If you want to see the characters you type in the hyperterminal, you must echo them back (or activate local echoing in the program).

Yes, I have the echo feature set in Hyper terminal and see the characters as I type them into hyper terminal. These lines are now commented out...

Can you describe again, what your problem is? Describe what works and what doesn't as well as what you expected.

OK, as you may recall from my earlier posts(s) that I have a control panel containing
60 plus buttons that is a serial device, I have this connected to my Arduino Uno
and the arduino connected to my PC's USB port. You mentioned earlier that this is a pass through type configuration because the USB and the serial port share the same ines (d0 = rxd, & d1 = txd).

I do have a level shifter between the arduino and the device to set the lines to rs232 levels and I can read data from the control panel when buttons are pressed.
When buttons on the contol panel are pressed, I can see a negative going series of
pulses on both d0 and d1 of the arduino. The d0 line's voltage level is between 0 volts
and ~ 4v while the d1 line's voltage level is between 0 volts and ~ 5v.

When I enter a key from my PC's keyboard,I can see a small negative going pulse that
appears at a voltage level of ~ 4v to a low of about ~3.5v and cannot see a pulse
on the txd line (D1).

In an attempt to isolate this to the ardunio, I have removed the level shifter part and tried this and I still am not able to see a pulse on the txd line (d1).

So the problem appears to be that I can not send data to the serial port control
panel device and the data that should be presented to d1 (from the USB port) is not...

Not sure what else to try at this point.