Date: 2/27/2008 5:00 pm EDT
You have posed one of the tuffer questions for Do It Yourselfers to overcome. The procedure is easy enough if you have the right tools. Most DIY'ers don't.
The larger hole for the stool flange can be cut by making a series of "plunge cuts" with your tile saw. Locate the position of the hole, draw it out with a Sharpie then go to nagging away at the tile until you have the hole satisfactory. This is better than cutting through the tile if the footprint of the fixture won't hide the cut. The hole doesn't have to be pretty.
The smaller holes are a little more difficult. You can again use your wet saw to grind away at the backside of the tile. You'll have to remove some of the plastic but only what is necessary. Once you have the tile thin enough from your grinding, you can then punch a hole from the topside and tool it nicely. Use a hammer and a screwdriver for example. You will have escutcheons to cover any missteps as long as they aren't too wild.
That is the more remedial way to do it.
A pro would use diamond coring saws (hole cutters) for the smaller holes and a right 4" angle grinder with a diamond blade for the larger stool hole. If you have access to those tools you're in good shape.
Avaire is a quality porcelain tile product which is good for the homeowner but hard to deal with when trying to tool it. To make matters worse Avaire is one of the thickest porcelain tiles on the market. Good when it comes to quality, bad when you have to get a hole through it.