Thanks for the code. Please edit your post with the code and use code tags
before the code
after the code
The result will be like
your code here
It's easier to read and copy and some more advantages.
OK, so we don't know how you're planning to connect your two Arduinos. I2C? Serial? SPI?
In it's simplest form, you can connect the TX pin of the Arduino with the SD card directly to the RX pin of the second Arduino and simply use Serial.read() on that one. What you currently see in the Serial monitor will also be send to the other Arduino. Don't forget to also connect a GND between the two boards.
Note that in that case you can not use the USB port on the second Arduino for communication with the PC (you probably even need to disconnect the wire between the two if you want to upload a sketch to the second Arduino).
Which Arduino's are you using? If you have ones with more than one Serial port (e.g. Mega), you can use the second serial port for communication between the two to prevent this 'problem'. I2C and SPI also don't cause this 'problem'. And lastly you can use SoftwareSerial (basically on any free pin).
Further you might want to design a form of protocol to indicate to the second Arduino e.g. the beginning and the end of the transmission. And the second Arduino might have to indicate to the first one that it has finished processing a bunch of data and is ready to receive a new lot.
first arduino sends 0x02 (STX) and waits for reply 0x06 (ACK)
seconds arduino replies with ACK; it now knows that a bunch of text data is coming
first arduino sends a bunch of data (e.g. a line) to second one and waits for ACK
second arduino sends ACK when done processing (sending over internet)
back to (3) if there is more data to read fro the file
first arduino sends 0x03 (ETX); second arduino now knows that the data is complete
See http://www.asciitable.com/ for the numbers
Use serial.write() to send STX, ETX and ACK.