Bubble Breeze Galaxy
Through The Poison Swamp
Another mini-game galaxy, and quite a fun one. Jump into the bubbles and the use the Wiimote to gently blow Mario around the level. The fans are useful rather than hazardous, just be sure to counteract their effect by blowing from the opposite direction so you don't go speeding into a mine. The moving obstacles complicate things slightly, but provided you move Mario around with frequent gentle gusts rather than constant huffing you should net another star with no trouble.
Buoy Base Galaxy
The Floating Fortress
Am I the only person who remembers Nebulus? Anyway, this one can be quite long-winded, and more than a little frustrating. Jump into the water and lure a torpedo into striking the large round weight in a cage. This frees the platforms you'll need, and sends them bobbing to the surface. Climb out, and to the left of the yellow flotation tanks you can wall kick your way onto the tower. Keep ascending, dodging Thwomps and bubbles, until you reach the cleaning robot. Spin the green valve to switch off the bubble blowing FLUDD look-alikes, and then use the Topman enemy to propel you to the next set of platforms, where a screw awaits. Spin the screw and you'll set the platforms in motion. Don't panic if you fall off - there's now a launch star next to the flotation devices which will return you to the cleaning robot. Navigate your way around the rotating platforms, collecting four pieces of a blue pull star along the way - and an extra life if you fancy. Defeat the enemy at the top for the last piece, then use the pull star to reach the sphere above. Leap over the red shockwaves, and make your way to the screw on top of this planet. Spin it down and the planet splits open. Now just dive down to the star and you're done.
BONUS: Green Power Star #1
The Secret of Buoy Base
There's another star lurking in this galaxy - a green power star that will get you on your way to unlocking the Trial Galaxies. It's quite easy to find as well. When you swim down underneath the Buoy Tower, you'll see a green pipe on the seabed enclosed in a cage. Lure a torpedo into smashing it, and drop through the pipe. You'll emerge on the underside of the galaxy. Now it's just a question of running across the pathway, avoiding the Bullet Bills and other hazards, then luring one of them into smashing the cage at the top. Bottom. Whatever. You've got a green power star. That's what matters.
Bowser Jr's Airship Armada
Grand Star #4
Sinking The Airships
Time to spank Bowser's kid once more. Defeat (or avoid) the enemies on each ship, jumping into the square black hole to activate the cannon. You're aiming for the flagpole - which can be tricky since it's possible to be a few pixels out and not notice. Luckily, there's a pull star at the top of each one, so you can grab onto that if you're going to overshoot. When you reach the ship with the Luma shop and Toad, cash in your star bits for more health (recommended) or an extra life and head over to the raft, squashing the Mandibug along the way. Spin the switch, and you're propelled in a straight line through an onslaught of cannonballs and dangling spiders. Jump over the cannonballs, and knock the spiders out of the way by shooting star bits, and you'll arrive at the end of the line. Launch star up to the top deck, and get ready for another boss fight. This one is fairly simple, but it can get hectic so keep your cool. You're lobbing green shells at Bowser Jr's airship again, but this time he'll retaliate with Bullet Bills and Magikoopas. Rather than dodging the Bullet Bills, you may find it easier to stomp the grating and drop down to the launch star for a few seconds while the air clears. Hit the airship three times, and he'll turn face on and start blasting waves of three fireballs at you. Hit the airship twice more and it'll explode, netting you the fourth Grand Star and access to the Bedroom.
BONUS: Drip Drop Galaxy
Feed the hungry Luma next to the Kitchen with 600 star bits to open this galaxy.
Giant Eel Outbreak
You need to kill all three giant eels to earn the star here, so you'd better be good at swimming. There are plenty of coins and air bubbles to keep you alive, but there are also lots of hazards - including homing torpedoes and spiky anemones. Grab a shell (red, if you can find it, green will do just fine) and swim towards an eel. It's best to wait until you're really close, since their swimming trajectory can carry them out of your line of sight after firing. Once you've got all three, the star appears on the shipwreck. Feel free to swim around and stock up on coins and other goodies - but be aware that the chests which can be smashed with the shells can contain a nasty surprise!