Mario's basic movement is accomplished with the analog stick on the Nunchuk attachment, which is exactly what you want in a 3D platformer. Small movements of the thumb start the little plumber walking and big movements propel him across the world quickly, yet accurately. Like the GameCube analog stick, there are eight notches to push into, which makes running from point A to B easier than if the stick were completely free moving. Mario has a quirk that simultaneously makes him easier and more satisfying to control: he can walk right to the edge of most platforms without falling. If he takes that extra step, he usually can grab the edge before falling thus saving a life for another day. It's rare to walk to your death in this game. (Running and jumping to your death commonly occurs, however.)
The other thing you want Mario to do all the time, of course, is jump and that control is conveniently located under your other thumb—the A button on the Wiimote. Again, this is exactly the way a platformer should be controlled. Pulling off precise jumps is made easier by giving the player the ability to change course midair, which is physically-impossible everywhere but in a video game. Jumping into a wall and jumping again in the opposite direction allows you to get to higher locations than would be ordinarily be possible. Many levels rely on the ability to climb parallel vertical walls and the maneuver is very satisfying to pull off.
A third command, which is commonly needed, duck was mapped to the Z button on the Nunchuk. Again, it would have been easy to put this on a shoulder button of the GameCube controller. Combined with movement, the duck becomes a duck walk and combined with a jump, it becomes an extra high back-flip. Tapping Z and A while running allows for long jumps and tapping Z in the air produces a ground pound needed to stun certain enemies. As you can see, there are a number of combinations that help Mario get from point A to point B. Pretty soon, you'll find yourself back-flipping into a wall jump and finishing off with a ground pound. And none of these controls use features not available on the GameCube.
The main action you need that is mapped to a Wiimote only control is Mario's spin attack. Flicking the Wiimote (or Nunchuk) in a vaguely circular motion causes Mario to violently spin with fists outstretched knocking around enemies. While there's no reason it couldn't be mapped to a button, the physical action of flipping the wrist maps well to the onscreen action of a spin attack. In addition, there's a short recharge period that needs to be observed before using spin attack again, so the relatively slower flick control matches the relatively slow development of the attack. As you might expect, the spin control is pressed into a variety of different uses in different portions of the game. You can get a speed boost underwater, turn giant screws, start skating on ice worlds, gain more jump hang time, spin a Boo out of the way and so on.
Galaxy makes moderate use of the IR controls to collect and fire star bits. Most levels are completely playable without this mechanic, but it does make some of them easier. Firing star bits takes a backseat to collecting them, since it's usually better to save them in order to get a new life. The second player essentially uses the IR controls exclusively. When I've been able to talk my son into playing, it's nice to be able to have one person dedicated to manipulating the star bit inventory. (The second player can also freeze most enemies, which simplifies many puzzles.) Other uses of the pointer include aiming the canon with Mario as ammunition, grabbing pull stars, positioning bubble puffers, etc.
It seems like the development team had some fun with some of the levels. For instance, several levels have stars trapped in transparent balls that Mario jumps on and are controlled by tilting the Wiimote. (Now where have I seen that before?) Another level lets Mario ride a giant swimming manta ray via tilt control. A great many puzzles depend on special costumes that alter the way Mario moves through the world. Fire Mario launches fire balls instead of the spin attack. Bee Mario flies a short distance in lieu of jumping. Boo Mario disappears when you flick the Wiimote. And so on.