To begin, in Blade & Soul the Hongmoon weapon and assorted accessories that you get at the start of the game is what your endgame weapon will be, at least for the game as it is released right now, and even beyond then, it'll be the weapon you'll level up with and evolve throughout the game. Weapons and accessories have two different systems to breakthrough, as the weapon will need a class specific breakthrough item (we'll go over this in a bit), whereas your accessories will need a generic item that's much easier to obtain (but their benefits are much less). Any weapons and accessories you obtain will be used to upgrade your hongmoon set, not to actually be equipped (or will be used for glamor, but that's another thing in of itself).
To start with, we'll go over the basics. The Hongmoon Sword, Staff, Razors, etc. you get at the very start is your endgame weapon. By level 20, you'll have received the necklace, earring, and ring as well. These items evolve, by combining weapons and accessories found out in the world into them. Once they reach their maximum level, you can "breakthrough" and evolve it into another weapon entirely, which will continue the process. You don't specifically have a choice of how it evolves, that's a fixed statistic, but you can further modify it with gems and the such.
Upgrading your Weapon and Accessory
To upgrade the weapon or accessory (I'll use the term gear for both after this point), click on it in your inventory and then choose "manage equipment." Here, in the center tab, you'll see a slot for "Amplifer." Drag a weapon for weapons, or any accessory for an accessory, into that box and then you'll see at the top a min and max on the level bar. This is how much at the very least it'll increase and how much at the very most it'll increase the experience (so to say) of the item. You can add additional ingredients to the mix, which will increase the amount of XP (useful when your'e adding a lot of weapons at once), which come in the form of fortune charms. You can also find gems that'll substitute for the amplifier (although do not use them if it'll max the item out, as you'll be wasting a good bit of it).
You can find the gear for upgrading as drops throughout the world and from the fortune wheels / world bosses that drop chests. You can find fortune charms from the hongmoon store and the daily dash, along with the replacement gems (which give huge XP bonuses). You can buy a lot of these items off of the marketplace too.
DO NOT UNSEAL REGULAR ITEMS AS ITS A WASTE OF A VALUABLE ITEM TO DO SO, ONLY FOR BREAKTHROUGH AND TRANSFORMATION
Breakthrough / Transformation
You can transform the item into a new item or breakthrough into the next stage by using a specific amplifier, along with a specific ingredient. For weapons, this is going to require RNG luck and you'll need to unseal it. These items will be obtained through generally either RNG box luck off of something like the fortune wheel or through a dungeon, which you'll need to RNG from defeating it. Accessories have the same requirement, but there it's much easier to obtain, since you don't have to roll for a chance to get a Blade Dancer's ring, as the ring will work for anyone. As for the extra ingredient, that'll come from quests, dungeons, or sometimes the daily dash, like the Viridian Transformation Stone.
You can lower RNG on chests by using a key from the hongmoon store, if you want to skip a lot of the farming.
You can change the appearance of ONLY your weapon, via the glamor system. It's the 3rd tab on the manage equipment screen.
There is no impact on stats as your gear levels up, however, you can modify the stats on the preview tab with gems.
There is no equipment in the game right now that is better than your Hongmoon set. Other servers use a "Stalker" set, instead, in which the Hongmoon set is used to upgrade, but the NA servers use the Hongmoon set, at least until additional content gets patched in.
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