Artifacts can be found through various ways like treasure chests, mission rewards, enemy/boss drops, or domains. Most enemy or bosses that don't require resins have a chance to drop an artifact when you defeat them, and they mostly drop the same few sets of artifacts, with a base rarity of 3 stars and a chance to drop 4 stars. The bosses that require resins, like Hypostasis, Regsivines, etc, drops a few other additional artifacts with a chance to drop a 5 star Gladiator or Wanderer's Troupe pieces.
Domains, on the other hand, drops unique artifact sets that are specific to certain domains. Domains are a dungeon like area that you can go to get specific artifacts, some can only be obtained from these places as well. However, it costs 20 Original Resin to be able to claim the rewards. You can do domains solo or multiplayer.
You can view the list of domains and their drops by going to your adventure handbook and selecting the "Domains" tab. You will need to unlock the domains first before you are able to see the it in-game though.
Added after the version 1.1 update, you can use Condensed Resin to pick up the rewards from domains. It costs 40 Original Resin to make 1 condensed, so using it on domains will give you double the rewards. Both sets of rewards are still randomized.
You can also farm artifacts without using resin because certain parts of the map drops them, and these resets every 24 hours based on when you picked them up. The artifacts that you get are mostly 1/2 stars, so it's mainly to be used as fodder exp for your main artifact.
The other way is to hunt down Elite enemies, which spawns every day. You can find them all easily by using any Genshin Impact interactive maps, but there are a couple of easily convenient spots with plenty of Elites shown on the page linked below.
There are only a set amount of these bosses roaming around per day, and they always spawn in the same location. Drop rates for these will increase the higher your world level gets. Use the Adventure Handbook and tap the "Navigate" on the bosses tab to find them easily.