Here is the full list of all 84 Little Friends: Puppy Island achievements.
Build a play area at the beach with your pups
Clear a teeter totter obstacle with a pup for the first time
Visit the Bountiful Beach for the first time
Befriend a chihuahua for the first time
Dig at dig spots on trails 25 times
Befriend a bulldog for the first time
Build a greenhouse in the gardens with your pups
Feed your pups 10 times at home
Befriend a dachshund for the first time
Befriend a shiba for the first time
Befriend and bring home 5 pups found on the trails
Befriend a pug for the first time
Clear a balance obstacle with a pup for the first time
Visit the Grand Gardens for the first time
Give a bath to your pups 10 times at home
Build a ranger station in the forest with your pups
Befriend a corgi for the first time
Travel 1 km on the trails with your pups
Clear a hopping obstacle with a pup for the first time
Dig at dig spots on trails 10 times
Befriend a dalmation for the first time
Play flying disc with a pup for the first time
Build a flying disc park in the forest with your pups
Build a shop in the forest with your pups
Clear a tunnel obstacle with a pup for the first time
Befriend a labrador for the first time
Play with your pup for the first time
Clear a jumping obstacle with a pup for the first time
Visit the Friendly Forest for the first time
Pet your pup for the first time
Befriend and bring home 10 pups found on the trails
Befriend and bring home 20 pups found on the trails
Give a bath to your pups 25 times at home
Give a bath to your pups 50 times at home
Acquire 10 clothing items for your pups
Acquire 25 clothing items for your pups
Acquire 50 clothing items for your pups
Dig at dig spots on trails 50 times
Travel 5 km on the trails with your pups
Travel 10 km on the trails with your pups
Feed your pups 25 times at home
Feed your pups 50 times at home
Build a B&B in the gardens with your pups
Build a chalet in the mountains with your pups
Build a base camp in the ruins with your pups
Build a lifeguard stand at the beach with your pups
Build a lighthouse at the riverway with your pups
Build a school in the ruins with your pups
Build a spa at the beach with your pups
Build a yoga area at the riverway with your pups
Take 5 pictures with your phone's camera
Take 10 pictures with your phone's camera
Take 20 pictures with your phone's camera
Complete training to level up your pups 5 times
Complete training to level up your pups 10 times
Complete training to level up your pups 20 times
Have all permanent jobs staffed by at least one of your pups at the same time
Activate DK mode for the first time
Level all of one pup's stats to their maximum
Scare away a pup while trying to befriend it on the trails
Have a full school with three pups receiving training at the same time
Level one of your pup's stats to its maximum through training
Have a full spa with three pups receiving pampering at the same time
Complete the main storyline
Win an easy flying disc tournament
Win a medium flying disc tournament
Win a hard flying disc tournament
Complete spa treatments with 2 of your pups
Complete spa treatments with 5 of your pups
Complete spa treatments with 10 of your pups
Find all secret statues everywhere
Find all secret statues at the beach
Find all secret statues in the forest
Find all secret statues in the gardens
Find all secret statues in the mountains
Find all secret statues at the riverway
Find all secret statues in the ruins
Completely fill the memory board at home
Acquire 10 toys for your pups
Acquire 20 toys for your pups
Acquire 30 toys for your pups
Visit the Majestic Mountains for the first time
Visit the Refreshing Riverway for the first time
Visit the Remarkable Ruins for the first time