There are three types of trees that you can plant in your town and on the island. This page explains what the types of trees are and how to get them.
The three types of trees are normal trees, pine trees, and palm trees.
To get normal trees, you can use a shovel to plant fruit or money. These will turn into the "normal" type of tree. Also, you can grow plain normal trees buy buying a normal tree sapling from the garden store. If you plant a fruit, the tree will grow three of those fruits. If you shake the tree, the fruits will fall, and in a few days there will be more fruit on the tree.
If you have a golden shovel and you use it to plant bells, there is a small chance that the tree will grow three bags of money.
On trees that are growing your native fruit, sometimes it will grow a different-looking fruit called a perfect fruit. These fruits are quite valuable and can be planted to create perfect fruit trees. Please read the perfect fruit guide for more details.
Once the garden store opens in your town, you can buy pine tree saplings there. Pine trees will only grow in the northern half of your town. If you try to plant a pine tree sapling in the southern half, it will wither.
To grow palm trees, you have to plant bananas or coconuts on the beach. If you plant bananas or coconuts in the grass, they will wither and not turn into trees.
To get bananas and coconuts, visit the island. When you first arrive on the island, you will be in a wooden building. If you go out of this building through the door in the south wall, you will arrive on a small island. There should be some palm trees here where you can get bananas or coconuts. To get other fruits, play minigames (island tours) and take some fruits. If the minigame doesn't require you to collect fruits, you can keep the fruits after the minigame is over.
To shake a tree, walk up to the tree trunk and press A. When you shake a tree, something might fall out of the tree. A fruit tree will drop its fruits. Also, a tree might have a bug hiding in it. If it's a spider or bagworm, the bug will come down on a strand of silk, then will climb back up after a moment. If you swing your net at the leaves of the tree (not hitting the trunk) you can catch the bug after it goes back up. You can also try to quickly switch to your net and catch the bug while it is below the leaves.
Trees that have nothing growing on them, when shaken, might drop a coin, a piece of furniture, or a beehive. If a beehive falls, the bees will chase after you. If you have a net, you can try to catch them. Here are some tips for catching bees. You can sell the beehive even if you don't catch the bees.
When you cut down a tree, it leaves a tree stump behind. You can dig it up with the shovel. Most tree stumps have a bulls-eye pattern, but sometimes a tree stump will have a rare pattern on it. There are benefits to leaving rare stumps in your town. Read more at the tree stump guide.