Better than nothing
Okay, this is obvious really, but some artifacts are better than no artifacts. When you first start the game you won't be thinking about them too much, especially since they're not tutorialised, but if you're early on, just equip whatever you have.
Each artifact has a few basic buffs and status effects. Some of them will just increase your attack or defence stats, either by a percentage or just with a static bump to the numbers, and you're better off with these than without.
However some of them are more specialised. If you're early on in the game, don't worry about this, but if you're at the point where you're thinking about ascending characters, then yeah, it's time to keep artifacts in mind more often.
The Artifacts for the job
Each artifact can be equipped to a character it is more suited to. You probably already know that characters in Genshin Impact have fairly basic job roles, like DPS, Healer, Support, and the artifacts can boost the properties of these job roles.
For example, the 4* artifact Martial Artist's Wine Cup offers a Cryo damage bonus - pretty pointless for characters which are not Cryo characters, honestly.
However these specific elemental effects only really come into play with rarer artifacts - you'll have three star and under artifacts for most of the early game, and they have more simple defence and HP buffs.
Special set bonuses
Things get far more interesting when you start looking at artifact set bonuses. Each artifact has a unique bonus which is applied if you have multiple artifacts equipped which belong to the same set - you can see which set they're a part of in the description of each artifact.
For example, the 2* Lucky Dog's Goblet has the Lucky Dog set bonus. If you have two artifacts equipped to a character with the same set bonus, you get a DEF increase of 100, and for four, you get HP restored whenever you collect money.
Again, these set bonuses can be curated to fit your gameplay style, and there are dozens of different sets to look at and choose from. In the early game it's not important to think about, but as you start mix-maxing in the late game, it becomes gradually more important. And again, these set bonuses can fit particular character roles, like Healers.
Zoom and enhance
Finally, it's no good just having a character in your party and not levelling them up. It's no good having a rare sword but not enhancing it. And of course, it's counter-intuitive to have a rare artifact equipped to your characters and not enhance it.
Enhancing weapons and artifacts is important. Again, as with every tip in this guide, this isn't something you should spend too much time focusing on when you first start playing the game, but as you progress it'll become more and more important.
If you've started focusing on set bonuses and making sure your characters have complementary artifact loadouts, then it's worth investing some of your items in enhancing those artifacts and making your characters as capable as they can be.