Upgrade artifacts in Genshin Impact by giving them other artifacts.
Just like weapons, Genshin Impact's artifacts are upgraded by scratching unused artifacts to gain experience. To do this, go to the Character screen, select the character whose artifacts you want to upgrade, and then select the artifact you have in mind. From there you will see the “Improve” button at the bottom right. Selecting this option will take you to the Upgrade menu, where you will notice the level and stats of the artifact listed on the top right.
Below is the “Upgrade Materials” segment. You have one of two choices here: “Auto Add” will fill all six slots with items corresponding to the selected rarity, although it can easily grab something you might want to keep. The other option is to manually add items until you have the desired experience or all six slots are occupied. You can mix and match these methods. Therefore, if “Auto Add” extracts something that you don't want to delete, you can replace it manually.
From there you will see the price of your upgrade in “gold” (Mora in Genshin Impact), and all you have to do is click “Upgrade” if you have the required part. The artifacts you selected will be destroyed and your artifact will be upgraded. Keep in mind that the higher level your Artifact, the better the items needed to effectively level the Artifact.
The maximum level of your artifact will be determined by its rarity. Thus, the artefacts of rank 1 are capped at level 4, then 8, 12, 16, 20 for each increasing rank. You'll want to focus on set artifacts that complement your character, so don't upgrade artifacts you don't want to keep unless you're completely behind the power curve. Considering how often artifacts drop around the world, this shouldn't be a problem.