You wake up on an island beach, with no memory of who you are. Getting off the beach is only the beginning of a dangerous journey, which will lead you on the trail of the people who tried to kill you.
Notes
The game features an interactive tutorial which is aimed at players new to adventure games.
The solution by Gunness has an error (it doesn't mention taking the hat from the man in the alley). I just uploaded the correct solution.
Thanks - I have updated the solution and added more comments to it, too.
Great game, by the way - the puzzles are simple but the atmosphere is quite well done.
Just finished the CPC version of the game. It was a great port. Highly recommended!