Note: You may right-click on Exits to instantly travel there, greatly speeding up the walking back & forth. This is a very useful piece of information that's easy to miss.

1- Use your Wallet on the window booth to get the map of the park.

2- Go to the duck pond. Click on the duckling to put him in the pond.

3- Go to the lake and speak to the gardener there. If you wish to know the translation to what he says, hover your mouse over the following text (warning: SPOILERS): "Sir, be careful, I saw a strange woman hiding in the bushes. She had a weird look on her face... she seemed crazy. Watch out, sir".

4- Go to the toolshed area by going around the right-hand side of the museum. Try to open the toolshed door. It's locked.

5- Go to the dome if you haven't already (as pointed out by a reader, you need to click the alcove on the other side). At this point your character should say he's explored the whole park and thinks he has been stood up. If you happened to pass through the dome on the way to the toolshed, the "stood up" part should have been said as you arrived at the toolshed.
Optional step for achievement: Right-click the alcove on the other side of the entrance to look through it.

6- Go back to the duck pond to see a disturbing scene unfold in front of your eyes, then use the Handkerchief in your inventory on the poor duckling.
P.S: Rumors say that naming your duckling "Pedro" unlocks a secret achievement.

7- Walk out of the area and back in to see a surprise box on the bench. Click to open it.

8- Go back to the lake and click the balloon note on the far left, then click the rucksack by the tree to get a Key.

9- Go back to the toolshed and use the key to open its door. Click the opened door to get a Telescopic Hoe.
Optional step for achievement: Go to the pigeons area to find out that they are now gone. Click on the stone head on the ground where the pigeons were to inspect it.
Optional step for another achievement: Return the key to the gardener's bag.

10- Go back to the lake and use the Telescopic Hoe on the balloon note to obtain it.
Optional step for achievement: put the Key back on the rucksack.

11- Go to the museum and click the statue on the right-hand side. You should be hearing music. Right-click the pipe below the statue to notice the music box is there.

12- Go around the museum through the right-hand side and click the device on the wall. Click the little strips until all three squares on the right-hand side are black, then click the green button. You should hear a noise.

13- Check the statue again. There should be water now. Take the floating music box to get another note from Catarina.
Optional step for achievement: right-click/look at all of the statues.

14- Go to the toolshed to find another box. Open it.

15- Go back to the entrance to see the ending of the game.
Optional step for achievement: Answer the call on the red payphone, located directly north of the entrance.

Note: There doesn't seem to be any way to access the museum and dome areas. I tried replaying the game without putting the duckling in the pond this time, but the story won't progress if I don't do that. So there do not seem to be any secrets or alternative endings for this game.