Goals are tasks to perform certain actions in the game, which are generally used to introduce game features to the player - completing them can award Experience, coins, diamonds or Treats, or some combination of the four. New goals to accomplish are usually added to the list of active goals after leveling up, but some goals require other goals to be completed before they will appear.
Actions taken to complete goals are not counted retroactively. For example, the goal "Buy a Flappy Flag" can and will only be completed if a Flappy Flag is bought after the goal has become active - Flappy Flags already on the island do not count. However, previous purchases can count towards Goals involving multiples of an item. The "Get 5 more large bakeries" goal is misleading because you achieve it as soon as you reach a total of at least 5 large bakeries, with at least one bought after the goal appears. You don't have to buy 5 more bakeries after the goal appears.
Goal requirements are counted across all islands - the goal "Hatch and place 3 new Bowgarts" will be completed even if two are hatched on Plant Island and one is hatched on Cold Island... or even if two are on Plant Island and one of those reaches Level 15 and is placed on Gold Island, with no other Bowgarts having been hatched.
The Goals menu can be accessed from the top-left corner of the screen, beneath the level indicator. New goals added to the list are shown by an exclamation mark symbol appearing in the top-left corner of the Goals button, and any completed goals are shown by a number appearing in the top-right corner. Goal rewards are not awarded automatically, but must be manually collected in the Goals menu.
Monster goals generally relate to purchasing and leveling monsters. They are the first goals encountered for a new player, with some of them forming part of the introductory tutorial.
Breeding goals are related to acquiring monsters with more than one element. It is by far the most common type of goal - among other goals, there are at least two goals relating to each of the twenty-five multi-element monsters.
Note that placing monsters on Gold Island can count towards goals for hatching/placing N monsters, such as the "Hatch and place 3 new Bowgarts" goal. Hatching monsters from the scratch ticket will also work.
Box all Plant Island monster species in a Wubbox on Gold Island
Place your 1st Wubbox on any island (except Ethereal)
Box all Cold Island monster species in a Wubbox on Gold Island
Box all Air Island monster species in a Wubbox on Gold Island
Powerup a Wubbox on Gold Island
Map goals relate to purchasing Islands. There are only two known goals of this type, one of which had arrived with introduction of Composer Island.
Get Cold Island
Get Composer Island
Tribal goals are unique to Tribal Island. Unlike the other goals, they are achived by the cumulative effort of everyone on tribal island, but the goal rewards are not given immediately. There is a time limit of one week in order for all of the goals to be achieved, and all members of a tribe will receive the reward when that time limit expires, at which point the goals are reset.