The first hint for each achievement is intended to nudge a player gently in the right direction. The last hint for each achievement is a pretty explicit explanation. Hints in between (if present) will be—you guessed it—somewhere in between cryptic and direct.

Community ranking of this guide is apparently only about 60% positive, which I find a bit surprising. If anyone has suggestions, leave ’em in the comments!

These achievements can be attained on the first playthrough.

Dying a lot is fine
hint 1 The play's the thing.
hint 2 Just keep playing until the ghost has caught Fidel several times.

Eat 4 Plants
hint 1 Step, step, squish and repeat.
hint 2 You'll need a fertile fourth level.
hint 3 Being able to “eat” 4 plants is a matter of chance. Even when 4 plants load on the same level, they may not all be gettable. Fortunately, this achievement is earned as soon as the fourth plant is squished, so you do not need to leave yourself an exit. Detonating a bomb may help in a pinch.

Overweight Rescue
hint 1 If you eat your veggies, you can have a snack.
hint 2 Secret rooms are for secret eating.
hint 3 Use a bomb to access a secret door in plant level. In the secret room, eat hot dogs until you're a thick pupper (you gotta eat ’em all). Then beat the rest of the playthrough.

Party is Over
hint 1 …only when the last guest has left.
hint 2 Squish every alien at the UFO raver.

Chew a Mimic
hint 1 It all hinges upon a good attack strategy.
hint 2 Chests can’t bite you from their hinge-side.
hint 3 Bark at chests to check if they contain gold or teeth. When you find one with teeth, attack it from behind.

Nobody dare touch my grandma
hint 1 Nobody may be an overstatement—grandma gets snatched a lot. Which grandma-snatcher do you have the most control over?
hint 2 Button masher? More like, button blaster!
hint 3 When the robot dog is about to snatch grandma, bomb the button instead of stepping on it. Exit without stepping on the button.

Secret Level
hint 1 Make 5 wishes!
hint 2 Blow out all the candles!
hint 3 When the dragon claw snatches grandma, extinguish the candles (ideally by barking) and enter the portal that opens over the pentagram.

Checkmate
hint 1 Pull a Ron Weasley.
hint 2 Defeat the chessmen in the secret level (see previous achievement).

Good Dog!
hint 1 Do the right thing.
hint 2 Save your grandma.
hint 3 Beat the first playthrough.

These achievements can be attained by playing the second playthrough (“Underworld”), which unlocks when the first playthrough is beaten.

No Zapping
hint 1 Don't be a Diamond Dog.
hint 2 Beat the Nightmare of Underworld 9 without using the gem-powered bark. Ideally, avoid collecting any gems in Underworld levels. If that fails, avoid barking while fighting the nightmare.

Fidel
hint 1 One small step for Fidel, one giant leap for mortalkind.
hint 2 Beat the second playthrough and crossover into the afterlife.

Dead dog CAN bark
hint 1 Even a good dog can’t do this trick!
hint 2 This isn’t accessible until the second playthrough is beaten.
hint 3 Bark sometime after you’ve crossed into the afterlife but before you reach Fidel’s little puppy grave.

These achievements can be attained when playing with the ball.

Flashback
hint 1 The best place to start a story is at the beginning.
hint 2 How can you interact with your ball if you can't reach it?
hint 3 Bark at the ball on the first screen, then start the first playthough.

BAD DOG!!!
hint 1 Passenger elevator or freight elevator?
hint 2 You'll need to bring your ball a long way!
hint 3 In the flashback, push the ball to the trapdoor. Then collect the ball in every level through the first playthough. When you have the chance to save grandma, save your ball instead. Try not to feel too guilty when grandma panics.

Gift from Beyond
hint 1 That rubber ball may outlast us all.
hint 2 You'll need to bring that ball to the end of this world and beyond.
hint 3 Play through every level of the Underworld, collecting the ball on each screen. After you defeat the nightmare and cross into the afterlife, push it into grandma's spectral hands.

These achievements can be attained once the mini-games unlock after Underworld is beaten for the first time.

King of Centipede
hint 1 100 feet/level × 9 levels ÷ 10 minutes = 90 feet/minute.
hint 2 Beat the Cenitpede minigame in under 10 minutes.

Zombie Rescue
hint 1 You'll need to start off on the right foot. Actually, make that 100 right feet.
hint 2 Beat the Cenitpede minigame and see what's new.
hint 3 Eat the worm sausage in the menu, then beat Levels 1–15 as Zombie Fidel.

Zombie Ending
hint 1 You'll need to start off on the right foot. Actually, make that 100 right feet.
hint 2 Beat the Cenitpede minigame, then make things difficult for yourself.
hint 3 Eat the worm sausage in the menu, then beat Underworld 1–9 as Zombie Fidel.

King of Puzzle
hint 1 XPs ASAP FTW.
hint 2 Beat the Puzzle minigame in under 20 minutes.

Robot Rescue
hint 1 *THE ONLY WINNING MOVE IS NOT TO PLAY*.
hint 2 Beat the Puzzle minigame and see what's new.
hint 3 On the menu level, eat the electric sausage, then beat Levels 1–15 as Robot Fidel.

Robot Ending
hint 1 *THE ONLY WINNING MOVE IS NOT TO PLAY*.
hint 2 Beat the Puzzle minigame, then make things difficult for yourself.
hint 3 On the menu level, eat the electric sausage, then beat Underworld 1–9 as Robot Fidel.

Not A Real Stair
hint 1 That ain’t no stairway to heaven. In fact, it’s quite the opposite.
hint 2 The nightmare doesn’t seem to be affected by bombs or worms.
hint 3 When the nightmare appears after you’ve put grandma in the elevator, use Robot Fidel to zap it before it can zap the stair case. Then try to climb the stair and fail. (Why will you fail? It's Not A Real Stair.)

Who you gonna call?
hint 1 Ghostbusters or Speed Racer?
hint 2 The best place to find ghosts is the Cemetery.
hint 3 When you encounter the muon trap in the Cemetery (the speed challenge whose entrance is located below the puzzle entrance), make one pass close to the trap (inevitably getting spiked once in the process) before you approach the button. After passing close to the button, spike yourself enough times to activate the ghost, then backtrack. When the ghost is within a one-tile range of the trap, press the button.

Daily Challenge XP Overdrive
hint 1 Bark on through to the other side.
hint 2 Beat the daily challenge with extra XP.
hint 3 The XP goal is optimized for a path that does not require barking. In rare instances, more optimal routes may be found by barking to reorient turtles or explode tiny spiders. (The usefulness of barking at tiny spiders is discussed in posts #19 and #20 of this thread.)

319 Bones
hint 1 See you tomorrow! And the day after! And the day…
hint 2 Beat the daily challenge 319 times.

A tip of the hat goes to Monty, glass, BarkingDoggy and ermagix for their contributions!