Rune Factory: A Fantasy Harvest Moon has eight caves in total. All must be cleared in order to complete the storyline. This can take as many years as you want it to take. You can only enter the caves in order and you need a permit for each one. In order to get the permit for the next cave, you have to have defeated the boss of the previous cave.
In Rune Factory each cave has a unique climate that is perfect for growing certain crops, like a green house in Harvest Moon: Island of Happiness. Each cave also contains certain monsters that are perfect for collecting, if they do something useful.
The monsters in each cave are coming from the Monster Generators. Ivan gives you a Monster Generator radar when you first enter Carmite, so you can always see where they are. You must destroy the Monster Generators so it will stop producing monsters. Also, the Monster Generator will always be back if you leave and reenter the cave.
In order to get through the caves you need to take your time and upgrade your weapons. A sword is a must!