Your 360 will be getting couriered away to the delightful people at the Microsoft Xbox Repair Centre. Mine's been twice and my flatmate's 3 times.
Here's what to watch for:
No hard drive or cables to go with it.
If you have discs in the drive, get them out. There's no guarantee of getting them back if you forget. I nearly lost Double Agent like that.
Faceplates. Take them off and put the white one on, as if they have trouble fixing your machine they'll send a new one back and typically they'll not bother with the faceplate.
You need to pack the machine up yourself for UPS, then attach the label they send you (it is possible to get the labels Emailed to you, but it sounds like your Customer service person was crap). You can track it to Microsoft via UPS' website if you keep a note of the Package number.
Then you wait 2-4 weeks to get it back. Sadly they won't be any more specific than that, even if you phone and hassle them. It'll come back in a custom fitted box tho, which is handy if you ever need to send it away again.