Using Uno w/ Dattalogging Shield to record Precision Barometer measurements

For that, you should only need 3 wires. Connect the ground to the arduino ground.

If they are correctly labelled, connect the Tx of the device to the Rx of the arduino, and the Rx of the device to the Tx of the arduino.

If you are only going to have the arduino listen to the device, and don't need to send any setup string or data, then you don't need the last one.

You also need to consider that some devices are labelled wrongly, and you actually might need to connect the tx to tx and the rx to rx.