I have some doubts about connecting it to Arduino.
This modem has both 12V serial output and TTL output. So I will use TTL output to connect it to Arduino...
However, when Arduino is connected to the PC, for debugging, can the pins 0 and 1 be used? should i connect the modem to 0 and 1 pins? I have to use a specific library for serial connection?
when I write serial.println and the Arduino is not connected to the PC, the communication goes through 0 and 1 pins?
However, when Arduino is connected to the PC, for debugging, can the pins 0 and 1 be used?
No. They are the serial port pins.
should i connect the modem to 0 and 1 pins?
Not if you want it to work.
I have to use a specific library for serial connection?
Depends on the Arduino. If you have a Mega, use another set of hardware serial pins, and use the corresponding SerialN class. If you have any other Arduino, use NewSoftSerial (for 0023 or earlier) or SoftwareSerial (for 1.0+). Do not use SoftwareSerial with 0023 or earlier.
when I write serial.println and the Arduino is not connected to the PC, the communication goes through 0 and 1 pins?
Serial data is transmitted by those pins, regardless of whether the Arduino is actually connected to the PC or not.
pins 0 and 1 do not work when the Arduino is connected to the computer, because then the serial communication is working through the USB port.
No, this is not right. Pins 0 and 1 DO work - they perform a specific function, though, so that they are not available for other purposes.
*if no additional library is used, pins 0 and 1 don not work for serial communication
No, this is not right. The HardwareSerial class, instanced as Serial, is what makes pins 0 and 1 work for serial communication. That library is part of the Arduino core.
As I am using 1.0 the library to employ is SoftwareSerial.
Yes, this is correct.
Then I will be able to communicate to the modem.
Well, that's a big leap, but, in theory, you should be able to.
But, OP wants to talk to the bluetooth module and the PC independently. So, either different hardware (Mega with multiple hardware serial ports) or different software (for software serial ports) is needed.