The tutorial is for testing, using the serial TX and RX is not very practical. Actually, it is wrong.
This is better:
Connect both the transmitter and receiver to your computer, and start the Arduino IDE twice. Select different usb-serial ports, so each Arduino IDE is talking to its own board.
Now you can develop both the code for transmitter and receiver, and can have Serial.println() on both, because you can also open two serial monitors.
Don't use pin RX and TX for the transmitter or receiver, use those only for the usb-serial port to the computer.
The only good software is VirtualWire or RadioHead.http://www.airspayce.com/mikem/arduino/VirtualWire/http://www.airspayce.com/mikem/arduino/RadioHead/
Connect a wire to the antenna pin. Any wire is better than nothing.
Check the wiring of the receiver careful, some receivers have other output levels or use a different pin order.