The cave is pretty much straight forward. Watch a show when you reach the end, Doga will tell you to acquire Noah's lute in Temple of Time which is your next destination. He then will teleport you outside.

1. Walk a few steps and watch a show. Proceed to the next room.
2. There is an Item and Magic shop inside. Fat Chocobo is the far right corner. In the next room, there is a pot where you can restore your HP/MP and a magic book on the table to revive your dead party members.
3. When you are ready, go to (3) through a hidden passageway but you will need to examine a switch first (candle). Now cast Mini spell to enter the Cave of the Circle, again, re-cast to reverse the effect after entered.