You have to:

• obtain the Wooden Bat to progress to the Duri Forest part of the plot
• gain access to Adar's House to get to Misty Gorge
• enter the mayor's palace to progress the plot beyond Atai City
• do everything plot-related from after freeing Lisa to entering Daea prison
• view the cutscene of freeing Thomas to progress to the Prince Tower part of the plot
• obtain the Artifact Shard from Leo to get to Dread Lands
• view the Rocket Boots cutscene to get the Artifact back

You don't have to:

• obtain any items except Wooden Bat, Slingshot, and Artifact (the Bombs and Rocket Boots are absent if you already have them)
• defeat Guard Gauntlet

Also, you cannot defeat the Duri Forest boss with explosives. (Disappointing, I know.)