There are two kinds of trees on the Island Farm - Fruit Tree and Wood Tree.
How do I get them?
There are two ways to get the trees. One is to unlock the plots on the Island to get the trees on those plots. The other way is to buy them in the Store. Go into the store on the island and check the Trees tab.
The amounts of trees you can have on the island are limited. For example, you can only have 5 Morus Trees on your Farm. The pricing will change for each tree (depending on how many you have).
You need Dev points (from helicopter orders) to unlock some of the trees (See FAQ: [Island] Dev Points).
Harvesting
You can get fruits from the Fruit Trees, and then use the fruits to produce other products in the machines.
From the Wood Tree, you can collect Wood. Tap on the ripe Wood Tree, and you will see two options: a Crosscut Saw (Craft in Hardware, chop 1 Wood per time) or an Electric Saw (click to buy with RC, chop 2 pieces of Woods per time) to collect it.
When you tap a wood tree, you can see the counter under its image (for example, 6/6 on the screenshot above). This counter means how many pieces of Wood you can collect from this tree. After you collect all wood from this tree, it will disappear.
When one Wood tree disappears, you can buy another one in the Store. If you have reached the limit, you won't be able to buy any of that tree from the Store. For example, if you already have 7 Oak Trees (reached the limit) on your Farm, no more Oak Trees can be bought.
Then you have collected 10 pieces of Wood from one tree, it will then disappear. At this point you can buy a new Tree.
You can use the Wood to produce other products in the Sawmill and Workshop (See FAQ: [Island] Buildings).