I have visited three of them, the Mausoleum of Halicarnassus, the Artemesion in Ephesus, and the Temple of Zeus at Olympia (where the statue was). They're all in ruins, with little remaining to hint at their previous grandeur. Still, there's some good evidence for what was once there.

In the Mausoleum's case, only a big pit and some nearby scattered stones remain. The nearby Bodrum Castle, was, however, built out of some of its remains in the 1400's. It seems to have already been a ruin by that time, so it was probably destroyed by earthquakes in the 1700 or so years since it was built. In the 1900's English archaeologists took away some of the remaining statues. Some of them, including a large statue of Mausolus, for whom the tomb was constructed and named, are on display in the British Museum.

Of the Artemesion even less remains. It's pretty sad, actually. It's literally just a big flat space with a single pillar and a few rocks. The first version of this temple was said to have been burned down by an arsonist named Herostratus. Fun fact! The Ephesians tried their best to wipe all knowledge of this man from history as punishment for his heinous crime, but only succeeded in making him more famous. This is may be the first recorded instance of the Streisand Effect. Afterward, the version considered a wonder of the world was built. It seems to have lasted until the 3rd century, when it was destroyed by Goths. In the next few centuries pieces of the temple were brought to other places to be reused. The nearby Basilica of St. John in Ephesus incorporated pieces of the ancient temple. Ironically, pieces of this now-ruined basilica were eventually incorporated into the Isa Bey Mosque by the Seljuks (which is located fewer than 100 meters from the basilica's ruins. Also, another fun fact! Isa Bey means "Lord Jesus" in Turkish, which means this is the only mosque in the world named after Jesus! It's not the same Jesus though, it's named after a different Jesus. Anyway, there are many remaining idiosyncratic cult statues of Artemis in the Ephesus Archaeological Museum in town. They're really cool and weird looking and they don't look like anything you'd imagine when thinking of Greek statues.

I guess the Statue of Zeus at Olympia has the least physical evidence of all, because it has completely disappeared. The temple in which it resided is still there, in a fragmentary state. Olympia was ravaged by many earthquakes and an intentional anti-pagan destruction wrought by Emperor Theodosius, and the statue seems to have disappeared at some point during late antiquity. We know what it looks like from artwork and coins, and we know it was constructed by the famous sculptor Phidias. In fact, his workshop was right next to the temple, and in it archaeologists have found Phidias' personal drinking cup. It's in the nearby archaeological museum and it says his name very clearly. Isn't that cool?

Anyway, I think we can safely assume these three, at least, existed. I can't speak to the others, but as far as I know you're correct in assuming only the "Hanging Gardens of Babylon" remain something of a mystery.

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Not to discourage any further answers but you'll probably enjoy this older answer:

Did the Hanging Gardens of Babylon actually exist, and has definitive proof been found? Why is it the only wonder whose location has not been definitely found? by /u/mythoplokos