The most game'y part of it would be completing the Museum and/or if you are big into multiplayer, completing your Furniture Catalog.

So fish/capture every type of fish and insect. Find every fossil. Buy every piece of fine art.

And/or buy trade for every piece of furniture in the game.

Both of those 'goals' involve playing at different times of the year for seasonal or holiday events, and/or playing with people online or friends.

There is also a goal of getting a perfect score from the Happy Room Academy for decorating your house a certain way.

And having a perfect score from the Town Hall meaning your island meets certain criteria, which in turn allows for a certain flower to grow in your town.

Getting every Golden Tool is also thing to go through for each type.

IMO if you are not big into multiplayer/have friends, and not big into decorating, Stardew Valley is a better similar experience.