Where to fly, and in what order
The world map lets you fly between locations in any order, but most dungeons are locked behind items you don't have yet. The chain is strict: each location gives you the item that unlocks the next.
Start in Duckburg. The Explorer gives Donald the Hero Key, and the level dead-ends at a wall too tall to climb. Fly to Mexico. Inside the Aztec Ruins, which is what the Hero Key opens, Goofy is wandering around with no agenda and gives Donald the Red Plunger plus a Strange Note that says "1 Sun 2 Moon 3 Star". Memorize that. It solves a puzzle several lands later, and the game does not remind you.
Back to Duckburg with the Red Plunger. The wall that blocked you the first time becomes climbable, and the rooftop route ends at Gyro Gearloose, who gives you the Bubble Gum Shooter. Gyro also refills its ammo for free any time you return empty, which is worth remembering for the back half of the game.
From here the map opens up. Transylvania is the next stop. Dracula is sitting on the full treasure map, and you cannot reach India, Egypt, the Viking ship, or the South Pole until you take it from him. The route through his castle needs the Bubble Gum Shooter to break colored blocks and create elevators.
With the full map, the order is Maharajah, then Egypt, then South Pole, then Viking Ghost Ship, then South Pole again. India gives you the Sphinx Tear after you beat the tiger. The Sphinx Tear opens Egypt's pyramid, and deep inside is the Riddle of the Sphinx room: the ceiling drops, and you have to step on the floor tiles in the order Goofy's note told you: Sun first, then Moon, then Star. Get it wrong and the ceiling crushes Donald.
The Sphinx room gives you the Scepter of Ra, which melts the ice block holding the Viking Key on your first South Pole visit. The Viking Key unlocks the ghost ship's lower deck; the captain there gives you the Green Plunger after you clear out the Viking Ghost. Then you go back to the South Pole one more time, use the Green Plunger to grab a passing bird, fly across the chasm, and find the Viking Diary frozen in the ice.
That's the last waypoint you control. Picking up the Diary triggers a cutscene with Pete, and the game takes the wheel from there, straight to the Ducky Gang Hideout and the final island.
How to beat the bosses
Every fight in QuackShot has the same trick: the boss is invincible most of the time, with one narrow window where damage lands. Once you learn the window, the fights collapse. Playing online in a browser means you can replay a boss room ten times in a row without restarting an emulator, which matters more for some of these than others.
Count Dracula (Transylvania, 5 hits)
Stand directly underneath Dracula and fire the Red Plunger straight up. He removes his cape and releases four bats, and the same upward shots that hit him also stun the bats before they can reach you. Start firing before he raises his arms, not after.
The Big Maharajah's Tiger (6 hits)
On the ground the tiger is invincible. He's only vulnerable mid-jump, so wait until he leaves the ground and fire upward while he's in the air. The Bubble Gum Shooter is the easiest weapon to land hits with. Stay in the dead center of the room. The fire platforms on the east and west walls breathe flame inward, so the corners are the worst place to camp.
The Viking Ghost (7 hits)
Sit at the far left of the arena. The Ghost won't attack you there. When he throws his axe, duck under it, then immediately shoot the Red Plunger at his head. After every hit he collapses and reassembles behind you, so sprint back to the left edge before he finishes putting himself together. Shots aimed anywhere below the head do nothing.
Big Bad Pete (10 hits)
Pete sinks into the floor and re-emerges inside a piston machine, a hydraulic press that tries to flatten Donald. The press itself is invulnerable. The only hittable target is Pete's head, sticking out of the top. There are two henchmen in the room who duck to dodge your plunger, so don't waste shots on them. Park yourself on the middle platform. When the machine rises near you, fire at Pete's head, then drop down and duck to avoid the henchmen's shots. Slide to dodge the piston when it comes back down, and climb back up for the next opening.
Guardian of the Great Duck Treasure (12 hits)
This is the longest fight in the game. The Guardian throws his sword across the room, where it embeds in the far wall; while the sword is stuck there, he's vulnerable. The catch is that stones fall from the ceiling at the same moment. Stand on the platform above the Guardian and slide right to dodge the falling blocks, then jump up and shoot his head before the sword returns to his hand. The instant the sword is back, he's invincible again. Climb to the upper platform and wait for the next throw.
Before you grab the Viking Diary
The Viking Diary is frozen deep in the ice on your second South Pole visit, and picking it up is a one-way door. Pete shows up in a cutscene, takes your nephews hostage, and the game auto-transitions to the Ducky Gang Hideout. You don't get a chance to fly back. From the Diary onward, you're committed to the final stretch: Ducky Gang Hideout, then Great Duck Treasure Island, then the credits.
This matters because the last two bosses (Pete and the Guardian) take 10 and 12 hits respectively, and the levels around them are stuffed with enemies you can't defeat, only stun. Showing up with empty Popcorn and Bubble Gum reserves means dragging the default plunger through the longest fights in the game. Before you head back to the South Pole, fly to Duckburg one more time. Gyro Gearloose refills the Bubble Gum Shooter for free as many times as you ask, so it costs nothing but the trip. Top up Popcorn ammo from any Corn pickups you've been saving.
A handful of enemies in the game can't be hurt at all, and trying to shoot them wastes ammo you're going to need. The list: bees in Mexico, caterpillars jumping from Duckburg dustbins, ghosts in Transylvania and on the Viking ship, Pete's diver henchmen surfacing from Egyptian quicksand, Pete's bulky henchmen who barrel across the deck and through the Hideout, and the killer whales jumping out of the South Pole water. Every shot you fire at them is wasted. Save the ammo for what you can hurt; you'll need every shot once you reach Pete.