USB or RX0/TX0 -> Serial.print
BEE or RX1/TX1 -> Serial1.print
When powering the board by USB you indeed don't need to connect the grove vcc line.
The output voltage of the USB is 5V, the grove and bee header have a voltage of 3.3V.
Both Arduino IDE, Putty or any other Serial Monitor will work.
Did you connect the grove pins to an ftdi chip before connecting it to your pc?