In addition to what Wierdy said, have you got any other guitars, especially with single coils? If so, do they exhibit the same buzz as the TLA? Just trying to exclude a common source of noise causing the buzz (like fluorescent lights or dimmer switches). Sitting too near the amp can often induce a hum. If you've only got humbuckers on other guitars, then a certain amount of noise pickup from single coils is common, but normally the level changes as you rotate the guitar (and can get to almost zero at the right angle).

Also, I wouldn't trust using the cheap guitar lead that comes with the kit, as they normally have very poor shielding. They work in a noise-free environment, but let in a lot of electrical noise if it's around. You don't need to spend a fortune on a guitar lead but anything new from say around A$20 should be OK.