You don't want to cut too many trees down, only the ones that are in the way of paths, or houses or that type of thing. For every tree you cut down, you should plant another.
You chop it with your axe and then dig up the stump with your shovel.
You can talk to Pelly about the environment to find out what needs to be done in the town, more trees or flowers, less weeds. Don't forget that the non-fruit trees are the ones with the bells and furniture, so you want plenty of those, too.
Plant the coconut like regular trees, near the sand but on the regular ground. Water them and they should grow. Lots of people seem to have a problem with the coconuts, though.
There are cool charms, too, the animals will tell you things like, "if a visitor cuts down a tree, your town will have good luck growing flowers" or something like that.
There is also feng shui for your house; put more green things on the south side, yellow on the west, and red on the east and you will have better luck getting furniture!
Unlike real life, the luck charms in the game really work!