List of Known Quests

Hot Spring Town

Guild Quests

Goblin Extermination (500G, 80EXP, Repeatable)
Delivery Work (300G, 50EXP)
Bar Waitress (500G, 30EXP, Repeatable)
Minotaur Extermination (5,000G, 250EXP)
Journey Bodyguard (1,000G, 150EXP)
Watching Children (1,000G, 150EXP)

When all 6 are done

Bandit's hideout (1500G+1500G, 150EXP)
Merchant Escort (2,500G, 500EXP)
Rescue Ashley (5,000G, 300EXP)
Goblin Queen Subjugation (800G, 100EXP)

Other quests

Recommendation for quest progression:
Babysitter → Bar Waitress → Delivery Work → Goblin Extermination → Minotaur Extermination → Journey Bodyguard → Bandit's Hideout → Rescue Ashley.

Until you complete the Rescue Ashley quest, only the West Gate of Hot Spring Town is always open for Nanako. After completing the Rescue Ashley quest, not only will most of the other gates become opened for Nanako (the South Gate will remain blocked until Nanako reaches Level 10), but you will also be able to ride the horse-train to and from the Capital for free. However, based on the prices of items and equipment at the shops of various towns, it is actually best to visit some of the other towns before traveling to the Capital.

If Nanako has reached Level 10, you can go south to visit Barqahasa. There's not much to do in Barqahasa events-wise, but you can visit the Armor Vendor to upgrade your equipment (including buying some semi-sheer pantyhose), and also rent a swimsuit and go to the beach. Then, simply use your Portable Portal to teleport back to Hot Spring Town. Next, head west to Grihastha to complete the quests there, and upgrade your equipment.

At this point, your character should be strong enough to complete the Merchant Escort quest, although you can also choose to complete it later, if you wish. It is not recommended that you attempt to complete the Goblin Queen Subjugation quest until much later, because the Goblin Queen is quite strong compared to bosses that you have encountered up to this point.