The gems consist of the Firefly Gem, Shiverchill Gem, Skywatch Gem, Bonfire Gem and the Water Gem, and another unknown gem which the Puppet Master had failed to show you. A new gem will be released this year (Astral Gem most likely), as Swoopy from The Lost Island says "Collect as much wood as you can before you real journey begins later this year". There has also been a conformation of a new gem from Van in the prodigy zendesk. Here is the photo:
As of this time, the Water Gem is unavailable due to changes being made to the Academy, as well as the rework of Barnacle Cove so it is like the other areas, where you move around into different areas, unlike Dyno Dig Oasis.
Therefore, you must collect the 4 gems already set out for you, wait to collect your Water Gem, and work hard to help Swoopy + the cottonweed fairies.