Trip with a monkey
First use Jack's compass to find the eye and the pistol. The first object is on the island on the right (dig it up using the spade lying nearby).  The second one is in the chests on the left. 
Take the cage with the monkey and put it on the raft.  Use the mechanism to sail the first part of the river. When the tentacles will wrap around jump on the platform with the boxes. Hit the one on the left and use the dynamite (the torch lies nearby). 
When you get to the second obstacle shoot at the three targets on water.  Grab the torch and jump to the left.  Remember that you can jump from the leaves (press A when you land on them). You can pick the torch from the shore (even when it is wet) so the last jump does not have to be perfect. Use the dynamite.
Repair the traction as Mr. Gibbs  and sail along. When you reach the other building you need to repair the helm. Get to it as Mr. Gibbs by going around the platform and jumping on the chests. 
Use the helm three times until the monkey will get on the platform.  Take the animal to the closest catapult. 
To fire the cage use Jack's sword to activate the mechanism near the machine. Shoot at the target that will appear above.  To travel along grab the rope which is lower (with the green bricks). 
Get across the bridges to the right and move the cage on the further platform. The monkey will be kidnapped by an alligator.  Jump left on the chests. 
On the left side of the river use Jack's hook to climb up.  Draw aside two chests. 
Run to the right across the narrow bridges.  Shoot at the target to lower the bridge. 
You will face the alligator. Dodge its attacks running to the left and right. Pay attention to what the natives throw at you - expect bombs. Pick them up and throw them at the alligator when it will show up with its jaws open.  Repeat the scheme three times. 
Use the pieces of the alligator to get on the island  Take the cage and walk the narrow plank.  Put the cage on the green floor panel.
The bridges will take you to the right side of the map. You have here two mechanisms which can be activated by the sword. Activate them and pick the winch that will fall from the net.  Go back and use it on the mechanism to lift another bridge. 
Roll the cage across the bridge  and put it in the mechanism that opens the door. 
In the hut
In the hut grab the two ropes to open the door upstairs.  Upstairs destroy the table's legs (on the right) to throw down the chest. 
Climb the wall and throw down all blue and white bricks.  Use them to build a path and move the chest. Activate the mechanism under the wall (using character with the sword). 
Go downstairs and assemble the pieces on the table. Later assemble the chair.  Use the spyglass. You need to find the ship in the south-west part of the map and follow it.