Pretty much any distro will do the things you want.
Ubuntu is the generally recommended noob distro. Vuze is actually built on Java, and will happily run in just about any environment, and Firefox has a Linux build that will run just fine (and ships with most distros already). Getting Nero to run in Linux is a pain because it's Windows software, but there are a number of free and open source alternatives you can try. Nautilus has a built-in CD/DVD burning tool, or if you want stand alone software
x-cd-roast is popular. For mp3's you have XMMS or Rhythmbox.
Caveat: Linux uses a different file system from Windows. You're going to have to format your hard drive. Linux can read NTFS partitions, but Windows won't know what to do with ext3.
There's really no such thing as a Linux distro that isn't advanced. Different distros pretty up the front end in different ways, but behind that it's all built on the same kernel. You can still do all the crazy console based stuff no matter what you're running (and will probably have to from time to time).
Wireless cards will generally auto-detect and install under Ubuntu. ATI provides binary drivers for their cards for Linux operation, although you may have to fiddle with them a bit to get them to work. Google is your best friend when it comes to error messages or general help on getting things to do what you want them to do.