mackrackit: Did you have a MAX232 or equivalent when you tried the hardware serial?
I don't use MAX232 for this circuit. I use a 22k resistor at Rx pin and 1k resistor at Tx pin. May I know is there any effect for not using MAX232? There is another question that I would like to know: can MAX3232 be used to replace MAX232?

Ioannis: What is the exact data sent from the SpO2 module? Is there any header that can be identified befor getting data?
The first byte of the 5 bytes data block starts with the synchronous bit of '1' while the other bytes start with bit '0'. This is the only identifier I could use to detect the the data bytes (SpO2 and pulse rate) I need.

I tried to check the bytes sent from the module since the starting of the module operation. The data might be correct for the first few bytes but the continuous bytes are all wrong.

My problem is similar to retransmitting a big file (in MB) through PIC. The file sent out from the PIC is not the same as the file before it was sent to the PIC. The PIC cannot handle a stream of continuous bytes and thus causing data went wrong.