I have put them on every concievable spot next to the lagoon, and no fishing! Is any spot better than others?

The villagers fish in the ocean, and it requires a clean beach and a certain tech level.

Huh? The page with hints to the puzzles I have bookmarked says they fish in the lagoon. Guess I should delete that bookmark, then! Thanks =)

I just checked, and it says the same in the guide on this site??

Which guide says that? I believe the only reference is that a strange fish can be caught in the lagoon. But only once, and you still need to have a certain tech level to be able to find it.

2 questions:
1: Are you trying to get the fish of Fertility? If so, you need a master farmer, and a certain level of tech.
2: Are you trying to fish normally? (by dropping a villager into the ocean) If so, then LadyCFII's post is correct.

I just want to fish for food. In the exact guide inhere in the Guides-section, it says they fish from the lagoon. It doesn't say anything about the kind of fish.

Yes, but the fish from the lagoon doesn't matter for now, to fish, you need to do what LadyCFII said.

Do I need to reach the 3rd level of farming to fish?

Yes.