OK. The blue chest is a bit complicated:
It requires you to be caught by the bandit lizards, and then to escape and retrieve your clothes. You'll get some "tatters" that you can take to the blacksmith... and then he'll eventually turn them into a new set of armour for you!
You'd have to go back to an old save in order to unlock this, though... so if you don't wanna go through all the hassle, I can say there'll be an alternative way to acquire this one in the future!
As for the green chest:
You need to be wearing the slimy black underwear, acquired by rescuing the blue-scaled ranger in the forest near the start of the game. So you may also need to go back to an earlier save.
Whilst wearing them, proceed to the East / right, past all the green slimes... until a scene occurs!
You can start a new game, or go back to an old save, and the scenes you unlock will be unlocked in your gallery. So don't worry, you don't need to play the entire game again.