You will start at Adamant Grotto, 1F. Walk and talk to the person guarding the entrance for an event, in which you can enter the grotto. Before you enter, make sure you have taken all the items in the treasure chests, which will help you by giving you a Tent, Potion, Phoenix Down, and Ether. Now, head downstairs to Adamant Grotto 1F. Biggs and Wedge will join your party.
After walking a few tiles up, there will be another event involving explanation about the moon phase, in which we are now at the Full Moon phase, which means that our physical attacks will be decreased by 25%, while Black Magic will increase at 25%. Definitely not good for Ceodore, being unable to learn any Black Magic. Thankfully, you have bought the Longsword and grind to level 10, and this dungeon will just be a breeze for you.
Walking further will get you into another tutorial, which involves Bands. Make sure you read this carefully, as you will need Bands to cause a lot of damage to your enemies. Continue on, while getting every treasure chest you see, which will contain an Ether, Potion, Tent to help you with your journey. Continuing along the way will lead you to Adamant Grotto B2.
Starting from this floor, Ceodore will be on his own, as Biggs and Wedge will leave the party. Walk North to find a Save Point, if you must, heal yourself using a Tent. Chances are its not necessary, however, so continue along the way.
You will see some stairs. From there, go west to get to a hidden passage, which leads to a chest containing a Phoenix Down. Return and follow the path east, go down another ladder to arrive at a chest containing a Bomb Fragment. South from there, you can see another chest containing Eye Drops. Before you left this floor, make sure that you are at level 12 at least. It will greatly help you.If you have, continue down the ladder to arrive at Adamant Grotto B3.
Go to the first chest you see on B3 to get to a chest containing a Bronze Hourglass. Go east across the bridge to see another chest, which contains a Tent. From there, return to original position and go south to obtain Antarctic Wind from a chest. Go west to see another chest which contains Echo Herbs. Go south, and go to the tiny path you see, to reveal a hidden passage to a new room. Inside the room, there will be Iron Gloves. Equip it. You'll need it. Return back and go to they yellow door, to arrive at a little room containing a Save Point. Save, then go down to B4.
This will be the final floor of Adamant Grotto. If you are ready and have saved the game, approach the chest to encounter the boss of this dungeon, Sand Worm.