Once you buy the shovel from Tom Nook's shop, you can bury items and dig up things from the ground.
Question: Where should you dig to find items
Answer: You might sometimes find these star shaped marks on the ground. Those indicate that an item is there and you can dig it up.
Question: What items can you dig up
Answer: You can find Pit Fall Seeds, Fossils you can donate to the museum, or maybe some furniture, etc. you or someone buried in the ground.
You can also bury Tree Saplings from Tom Nook's store and they will eventually grow into trees. (Remember to take care of them!)
Also, you can bury the fruits that grow around your town and they will eventually grow into a(n) (Fruit's Name) Tree.
Try to bury other fruits not native to your town.
Example: If the trees in your town grow Oranges, get a different type of fruit from another town and bury it in the ground. When it grows into a tree, shake off the fruit and sell them to Tom Nook. You should get 500 Bells for each of them!
The fossils you may dig up can be identified by the brown owl in your town's museum. When he identifies the fossil, you have the choice of donating it to the museum of keeping the fossil. If you do choose to keep it, you can sell it to Tom Nook for some bells.