You can upgrade Cecil's Light Sword later in the game to a more powerful version called Excalibur for a little travelling work. You must go to the Land of Summoned Monsters and retrieve a specific item and do some trading. You do not have to complete this quest as you eventually get a better sword, but it is a good sword to have if you get it as soon as you can.
The adamant armor is the most powerful body armor in the game and it can be worn by any of your characters. Unfortunately, you cannot begin to start the process of getting it until you are almost at the end of the game, in the Moon's Core. You also require a large amount of luck and patience. You need to fight a particular random enemy in a specific room of the Moon. If you are lucky enough to get the right enemy, you then need to be even luckier and hope they drop a special item called a Pink Tail.