To get rare materials in New World, you will first need to raise your skills at gathering materials. Once your skill is high enough, you will notice more and more rare materials as you gather in the game.
However, this will not be enough on its own to get rare materials as they will still be, as the name suggests, rare.
Because of this, you will need to try and get mining luck and the basic luck bonus. Both of these can increase the rate at which you gather rare materials.
You can get Mining luck from perks on tools, but depending on the tool, it will be different. For example, pickaxes have mining luck, and sickles have harvesting luck, and so on.
Due to this, you will want to create quite a few tools, until you get one with the luck perk on it. Using the tool with the perk will then increase the chances of getting rare materials.
Related: How to make dye in New World?
But for the basic luck bonus, you will need to join a faction, and you should get a bonus under territory bonuses in your bio shown below.
If your faction owns territory, you will get these buffs. Because of this, it would be ideal for joining the faction on your server that holds the most territory.
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