The prologue quest is a short introduction/tutorial for Dragon's Dogma, that will let you learn the controls, combat system, and other basics of the game. You start as Savan, a fighter. Follow the onscreen prompts.

You should see an update to your quest log:

Quest Log Updated!
The End at the Beginning
Pursue the Dragon

The first tip will show you how to use a lantern. Using a Lantern will hold darkness at bay. Open your inventory, and find this item:

Lantern
A small light source that burns liquid fuel, which might be replenished. It can be worn on the waist. Has enough oil to burn for a while.

Use the Lantern. Then, follow your companion, Salde, though the tunnels (or you could just run ahead). Soon you see the dragon, who shoots fire at you. Keep running, kill two goblins that block the path to Dragon's Dogma. You can practice heavy attacks and light attacks against them.

Pay heed to the world around you. Some ledges may be scaled, but a long fall will end in injury - or death. Soon you come across a Riftstone. Invoking this stone can gain you new allies known as "pawns", whose loyalty in battle will never falter. Invoke the etched stone to summon pawns from beyond. As you progress in your adventures, pawns will provide you with useful information and hints. The onscreen tips will tell you how to issue basic orders to your pawn allies and how to perform weapon skills. Pawns who join you are Quince and Morganna.

Continue through the tunnels. You reach a chamber where you fight a hobgoblin and some more goblins. Kill them. Continue south. Dash up the stairs, where you fight a Snow Harpy. Enter the Temple Antechamber. You can pick up an explosive barrel and throw it onto the group of enemies below. After it explores, charge down and kill any remaining enemies. There is another battle against Snow Harpies.

The gate to the south opens. Enter through it and follow the other survivors. You will now fight the Chimera. The hints on the screen will tell you how to grab on to and climb upon larger monsters, at some cost to your stamina. Try grabbing on to the Chimera's tail, then while you are clinged to the creature, slash at it with light attacks. Focus on the Serpent, and then on the Goat's head. Make sure to get healed when needed. Keep hacking on the Chimera as it stumbles on the ground. Attack the Lion's Head. This will be a pretty long battle. When you defeat the Chimera, the quest log will update the quest status to successful/completed, and now proceeds to some cinematics, followed by character creation.