Looking at specs online, your Asus has decent onboard sound and should be fine.
The headset has a green stereo connector (which connects to the green audio output of the computer) and a pink connector which is for the mic and connects to the pink output on the computer. The only thing odd there is the pink connector is a multi-channel (or stereo) connector whereas most are mono. What does the booklet say about that?
You should be able to just use the green connection for now just to get it working. Have you also gone into Windows audio properties to make sure that audio output is enabled when the headset is plugged in? If you're still having trouble, can you try the headset on another device or computer?