You may only have 3 artifacts equipped at any time. Those are the only 3 that have any effect. Some artifacts are Passive which means their effects always apply while equipped, and others must be activated and then have a cooldown until you can activate them again.
To change the three artifacts you have equipped, open the Advancements tab and click View All next to artifacts. Then drag the one you want to equip into a box at the top.
To activate a non-passive artifact, equip it and then look for the Activate button on the advancements tab. It will then "cool down" until it can be used again.
Note that you cannot un-equip an artifact that's cooling down. You also cannot equip two identical artifacts.
 Getting Artifacts
Artifacts can be found by digging at dig sites that you find while playing Warzone Idle. All Warzone Idle levels have dig sites other than the tutorial, the first level, and the bonus level.
Once started, a dig site disappears from the map and will be shown on the Advancements tab.
Note that the times specified in dig sites, as well as artifact cooldown and upgrade time, are in real-world time and are not affected by how often you open the game or what your maximum Idle Time is set to.