One or two players can play this mode. There is one floor in the game that requires two players to clear properly, but the game can still be beaten alone.
Choose New Game. Choose the character you want to play as and hit Start. Give your character a name. Remember this name! It is crucial if you wish to continue with a password later. If you just hit Start you'll be given the default name (Thor for the Warrior, Thyra for the Valkyrie, Questor for the Elf, Merlin for the Wizard).
You appear in front of the Magical Room (enter this to quit or modify your password). Head up and touch any of the men. Despite being thieves in the Arcade Mode, in Quest Mode these characters are instead merchants, and will buy and sell equipment. He will see you are an adventurer and gives you some advice as well as a Heal Drink and a Warp Ring. The Heal Drink heals you while the Warp Wing returns you to the entrance area. Note that both items vanish after each use, so use them wisely.
You can now head into any of the four towers. This guide starts with the Fire Tower.