Dragon's Dogma Walkthrough Part 13: Deep Trouble, Extermination, Rat Catcher

When returning to Cassardis after the events from the Encampment, there is a cutscene. Watch the dialog between the Arisen and the mysterious hooded man. Then, talk to Poll who is standing near the entrance, next to the well. He should have a green question mark above his head.

Poll: Gives me chills just to think of it... Ah! Er, I was- I didn't...! I'll make no excuse, cousin. I hid in the well for fear of the Dragon. I was too craven to venture far, and lucky thing... a queer growl echoed back with the sound of water from further in. There's aught down there, mark my words. I could feel it, I could.You'll not see me going back down there to find out what!

You have two options:
Investigate the Well
Not now

choose the first option to start the quest:

Deep Trouble

Objective
Unnerving snarls from Cassardis' old well startled a soldier seeking refuge inside. Investigate the source of the sounds

Details
Investigate the Well
Enter the well in Cassardis and explore its depths

Poll gives you a lantern to help you. You probably already have one, in which case you should just give the lantern to your pawn. But before descending into the well, you should check the Cassardis Notice Board at the inn since there are more quests you can complete while down in the Well, including: (Successful completion of each earns a base reward of 500 gold and 500 experience)

Rat Catcher

Details - Objectives
Slay 7 mice
Region: Unknown

Reward: Experience (800)

Request: If there is one thing I cannot abide, it's mice!

Extermination (this one is available only after you've killed the Hydra in the Encampment)

Details - Objectives
Slay 24 giant bats
Region: All Gransys

Reward: 50,000 G

Request: Every year, more and more animals fall victim to monster assaults...

Interact with the well to climb down. The cave you enter to is called "Dripstone Cave". It will be dark down here, so be sure to open your inventory and use the lantern.

Rat Catcher: There should be plenty of rats wandering the caverns. You should have no problem finding seven of them to slay. Once you kill 7 rats, you complete the quest Rat Catcher

Extermination: There are also some bats flying around the cave. It's possible though, that you do not find all 24 bats in one run. You can come back after a few days when the bats have respawned, or you could try hunting during the nights in the forests near Cassardis, or you could go to other dungeons and areas such as Everfall or Ancient Quarry. Slay 24 bats, and you will complete the quest Extermination.

Deep Trouble: Explore Deepstone Caves. When you read a dead end (with a chest near it, possibly a coin pouch inside), there will be a ledge on the side you could climb onto (the pawns might give you a hint regarding this). Keep climbing, then go down the tunnels and reach another dead end, but you can jump down to the stream of water. There, the quest will update to "Slay the Monsters". Attack and kill the Saurians, which are Lizard-like monsters.

The pawns might provide some strategies and tips. The scales along their backs are hardest of all. Their weakness is their tail, which you should try and sever. Otherwise, they will block all the frontal assaults. The scales atop their heads are soft. Slay them one by one. Their spittle is fell poison, so watch out for that. If you are having difficulties beating them, you can always come back later when you have leveled up and have better equipment.

Once you have killed all the lizards, the ques updates to "Report back to Poll". You can gather some Purple Anise in this area, near the waterfall, before heading out. Climb up another ledge and go through a long tunnel until you reach a ladder. Climb up the ladder to exit Dripstone Cave. Talk to Poll to complete the Deep Trouble quest and receive your rewards of Gold, Experience, and Rift Crystals.