Animal Crossing is an odd game. You do (generally) whatever you feel like.
The first goal you are given in each game is to pay off the mortgage for your house. You can find various ways to earn enough Bells (the in-game currency system) to make payments - you often start out picking up and selling seashells, but can also look into growing and harvesting fruit, catching rare fish or bugs to sell, or a handful of other tasks. The trick here is that every time you pay off one mortgage, your house is automatically upgraded and a new debt given to you.
With a growing house, you'll probably want to collect different types of furniture, wallpaper and flooring, and other accessories to decorate each room. You can get furniture by shopping at Tom Nook's store, or by trading letters with friends, or by shaking trees, or somethings through other methods.
Oh, and don't forget to cultivate friendships with other creatures in your town. Some you'll probably like, others not so much. It's up to you meet each different personality and decide what kind of relationship you want to have.
Other common goals in the game are to collect different things for the museum. There are exhibits for fish, bugs, art, and fossils. Completing each exhibit is a long-term goal that will often require you to play over the course of multiple months (since some fish and bugs are only available in different seasons, for example).
And of course there are other customizations you can do. Help to decorate the town, change the exterior of your house, design new clothes to wear, etc etc.
There are no hard goals you need to complete - it's really about doing whatever you feel is most interesting. Become a Bell millionaire, or own the biggest house in town, or decorate your world to perfectly reflect your own personality. If you get bored, you can also play with a friend and exchange messages, help build something together, or find hilarious ways to troll them.