Weapons can be upgraded in 2 ways: Leveling them up, or refine them. Leveling them will simply increase their current stats, and also the substat at every 5 weapon levels, while refining them is to increase their weapon effect.
Equipments that are at least 3 star rarity have extra effects on them. This effect can be enhanced further if you Refine them, and you do this by feeding them duplicates of the same weapon. The maximum refine rank is 5, so you need 4 other duplicates of the same weapon to max it out.
Similar to character progression, you can level up and ascend your weapons. The main difference is that you can feed it any weapon if you want exp, so you don't have to strictly use weapon exp materials, which are the enhancement ores. Once you level them out to their current max, you will need to ascend them.
You can get enhancement ores from various things, but the easiest way is to find mining rocks out in the world, then go to the blacksmith in town, and craft some. Just keep in mind that some of the later rock chunks can be used to craft some weapons, so you might want to consider that before using them all up for enhancement ores. It's also not recommended to over invest in 3 star weapons, since you can get 4 star weapons from gacha, exploration, or crafting.
You can simply think of Artifacts as "gears", and there are 5 different pieces that you can equip: Flower, Feather, Hourglass, Cup, and Crown.
Artifacts have a main stat, which is the main number at the top, and substats, the ones below the stars. Flower artifacts will always have flat HP as the main stat, and feather will always have flat ATK, but the other pieces can have different main stats which will be explained more below. Equipping artifact sets can get you extra bonuses as well.
Every 4 levels of an artifact, you can gain a substat, up to a maximum of 4. If you already have 4, then every 4 levels you get will upgrade one of the existing substat instead.
Artifact substats are kind of random in terms of the number of substats, and also the value of each of them, and so the same gear can have different substats.
You can easily view your available equipment sets by going to the "set" tab when you are in the equip artifact screen. This can make it a lot easier to know what to go for using what you have. Early on it might be confusing to know what to go for, but you will start to understand it the more you play the game. If you're unsure of what to use, then you can't go wrong with just raising HP and ATK, unless your character is a tank/healer, then you'll want to equip artifacts and sets that increase HP and DEF instead.
Base # of Sub Stats
Generally recommended to start with rarity 3 since it's guaranteed at least 1 substat(s). Early on, you don't really need to get the best possible rolls on them before levelling since it can be difficult to get a "perfect" roll. You will start to get better ones later when you want to farm them from domains.
Possible Primary Attribute
Base HP % / Base ATK % / Base DEF % / Elemental Mastery / Energy Recharge %
Base HP % / Base ATK % / Base DEF % / (Anemo/Pyro/Cryo/Electro/Hydro/Geo) DMG Bonus % / Physical DMG Bonus %
Base HP % / Base ATK % / Base DEF % / Elemental Mastery / CRIT DMG % / CRIT Rate % / Healing Bonus %
Percentage substats are only based on a character's base stats + character's weapon atk, which you can view by going to your character section and tap the "details", not on the artifacts section. This means that at early levels, it's better to go for flat numbers instead of percentage increasing stats since your base stats are low.
What stat to go for?
It's generally recommended to go for artifacts with ATK main stats and substats, because most damage dealers need higher ATK to deal more damage, even for characters like Lisa that attacks with an element, unless you are raising a support type character like Barbara, then you will want to go for HP/DEF main or substats. Crit rate is also not a bad option if you happen to get them, but it's usually only start becoming worth it once you have high crit damage to go along with it. Elemental mastery is only good for characters that can consistently proc an elemental reaction since it raises those damage instead.