Here is the full list of all 31 Let's Pet Pets achievements.
Get your total pet level to 250 or higher
Hatch 50 or more pets
Earn over 100,000,000,000 pet points
Get your total pet level to 100 or higher
Earn over 3125 heart points
Get your total pet level to 50 or higher
Hatch 10 or more pets
Hatch 25 or more pets
Earn over 100,000,000 pet points
Earn over 625 heart points
Earn 100 Pet Points from Petting
Hatch your first pet
Earn 5 Heart Points Total
Prestige for the first time
Upgrade the Farmer to level 10
Upgrade the Stylist to level 10
Upgrade the Nutritionist to level 10
Hatch 5 or more pets
Earn over 100,000 pet points
Earn over 25 heart points
Earn over 125 heart points
Get your total pet level to 500 or higher
Get your total pet level to 1000 or higher
Earn over 100,000,000,000,000 pet points
Earn over 100,000,000,000,000,000 pet points
Earn over 100,000,000,000,000,000,000 pet points
Have 50 total heart levels
Have 100 total heart levels
Have 250 total heart levels
Have 500 total heart levels
Have 1000 total heart levels