All weapons and accessories only have to have been in your inventory at some point during the game. You do not have to keep them unless they upgrade into something else you need for the achievement (Aqua Ring -> Riptide -> Nereid, for example). Once you think you have/had all the items, speak to Bahkti. (refer to Pulsian Pioneer for the Bahkti mini-quest)
Also, you're not required to upgrade every weapon to it's tier 3 form. Instead, upgrade them all to tier 2 and pick just one for each character that you'd like to be tier 3 (eg: Lightning's Omega Weapon). This will save you a lot of time and money.
The elemental rings can only be obtained two ways; one is from finding them in treasure spheres, the other is dismantling their respective charm. Be warned, if you sell your rings and charms, you have void yourself of this achievement.
The following missions will award you with certain elemental rings.
• Mission 06 - Fulmen ring (tier 2 Spark)
• Mission 11 - Frost ring
• Mission 27 - Blaze ring (tier 2 Ember)
• Mission 39 - Siltstone ring (tier 2 Clay)
• Mission 52 - Gale ring (tier 2 Zephyr)
• Mission 53 - Blaze ring (tier 2 Ember)
• Sulyya Springs [Treasure Sphere] - Riptide ring (tier 2 Aqua)
All the charms are found in treasure spheres on the Archlyte Steppe.