After the first dungeon, you can explore the western half of the island, and visit the second and third dungeons.
Your progress will be limited by how many dungeon treasures you have found. Each treasure gives you the ability to get past a new type of obstacle. Since there are overworld treasures to find, it's helpful to map out as much of the island as you can reach, then when you can't get any farther, go to the next dungeon to get the next treasure.
There are plenty of helpful items and features to discover as you explore. Below are some tips for finding them.
Be sure to go into buildings that you find. Doing so will add that location as a bicycle warp point on your map.
If you're ever stuck, or forget what to do next, check the Clues menu. It has a list of things to investigate.
At the Shadow Temple near the Elder's Cave, there is an altar with a diamond-shaped slot in it. Now that you have the Power of Fortitude, you can put it into the slot to unlock multiplayer.
Clear All Enemies
On some screens, if you kill all the monsters on the screen, items will fall from the sky afterward, such as potions, tunics, meteors, and more. These are all helpful items to have, so it's recommended to clear all enemies from every screen, both in the overworld and in dungeons.
Talk to NPCs
You may find NPCs standing around, especially near where you start the game. Most of them just have random things to say, but some will give you items, so be sure to talk to them.
If you find a ladder that goes down into a hole, or a cave opening, you can go in to find a mini-dungeon. These are smaller than the main dungeons and can contain random treasure, random NPCs that give you items, random merchants, ladders to other areas, as well as a mini-boss, which rewards meteors and weapon blueprints for the blacksmith, among other things.
As an added bonus, once you enter a mini-dungeon, you will be able to use the bicycle to warp to the map square where you found it.
As you explore, you may find potions. Their effects are unknown until you use them. After using a potion once, you will be able to identify that type of potion from then on. Note that some potions are dangerous, so you might not want to use them if you are low on health.
The Phoenix Ash is a rare item that can be found at random. If you get this item, check on Henrietta to evolve her into MeggaHenrietta. She will no longer hide when enemies are on screen, instead attacking them. She becomes covered in flames, which will damage enemies and melt ice. This can allow you to do some sequence breaking, by maneuvering MeggaHenrietta to touch ice that is blocking your path, since you normally can't melt ice without a dungeon treasure.
Bruce and Gourdon
You may find a bat that doesn't attack you, or a pumpkin that you can't pick up. Check on either of these, and if you say yes when prompted, you can have one of them follow you as your pet instead of Henrietta.
There is an item trading sequence that starts with a thirsty person. You will need to find a water potion and give it to them. They can identify the water potion even if you haven't identified it yourself yet, so you might want to hold onto potions so that you can give it to the thirsty person.
Library books can be found randomly. You can call Paige after each book you find, to hear a library pun. After you have found 8 library books, you will get a call from Paige, asking if you will visit the library. Go talk to her and you can have her join you, replacing your current pet. (You can switch back to your previous pet by talking to it in the library.) Paige doesn't hide when monsters are on screen unless her health is low. She throws books at enemies.
Rare Hidden Items
In dungeons, it's possible to find secret weak walls by walking up next to them and seeing them shake. If you have bombs or a hammer, you can break such walls to find a secret room that contains a rare item.
Some gravestones can be pushed downward. In 32-bit graphics mode, gravestones that can be pushed downward have a down triangle on them. In 8-bit, the gravestones that can be pushed are rounded on top. Sometimes an enemy will pop out, sometimes an item, or sometimes a ladder to a mini-dungeon will be revealed.
Second and Third Dungeons
The second dungeon will be marked on your map after you complete the first. Go there and get the treasure, optionally kill the boss, and leave. You'll get a call from Prince Daniel. You can use the treasure you got in the second dungeon to explore more of the map and reach the third dungeon. Get the treasure there, optionally kill the boss, and leave. After the third dungeon your next destination, Guild LLC, will be marked on the map.