The gearscore is calculated as average between your expertise and the actual gearsvore of the piece of equipment. So buying 590+ stuff was a good idea to get you a strong start.
The expertise will increase whenever you drop or loot a piece that has a higher gs than your current expertise in that category.
You can increase your gearscore through different means:
World Tours/Elite Chest runs - a group of players coming together to raid the elite areas
Lvl 60+ Expeditions - Lazarus, Genesis, Tempest’s Heart, Barnacles & Blackpowder (up to 25 runs per day). Here you can also get up to 2 Sapphire Gypsum per day.
Gypsum Orbs - every day you can earn a multitude of gypsum through various activities in Aeternum. These gypsum can be traded into gypsum orbs at gypsum ovens (Reekwater & Ebonscale cities have one aswell as the outposts in neutral territories). Craft the orbs into casts and open those to bump your gearscore for the selected category.