The Dragon Cave is a cave surrounded by several Amythite and Amythite covered objects. You get to explore and use this after completing the quest 'Into The Gloom' and may move it by completing 'The Dragon Cave' questline. To use it and it's features you must first clear the four Amythite rocks and the three trees in front of the mouth of the entrance.
Using The Dragon CaveAfter completing the quest 'Bump In The Road' you finally have the ability to search it at the start of 'Shrouded In Mystery'. To keep exploring after your two searches you must complete 'Faugrimm's Scepter Parts Machine' to keep exploring the cave. After completing that quest and the quest after you have the ability to move the cave anywhere.
You must have a sufficient amount of Mythium to keep exploring. If you happen to run out you can buy Amythite trees, rocks, etc. from the store to mine.