To look for meteorites you should be a Stargazer (or later Evolution) and leave the estate and go to 33, 58 … if you have emptied your starting chest you should have gained a key (this is invisible in your inventory) to the Springlade Observatory.
Enter the Observatory and either speak to the Celestial Supplier or click on the Star Seeker item on the ground to gain your first Star Seeker tool, this is what Stargazers use to find the meteorites.
You should use the Character > Shortcusts screen and create a shortcut your seeker so that you can use this ability easier.
Using the Star Seeker will show a tracking map letting you locate a meteorite in the current realm, you can then click on the meteorite to travel to its location and pick it up off the floor. However you may only carry one meteorite at a time and the Star Seeker will not work for you properly if you are carrying one as the proximity will make it go wonky.
Take meteorites back to the NPCs at 33, 58 in Knotwood Valley and hand them over for new weapons and other items.