At the start of your journey as Agent 3 in Splatoon 3's campaign, you'll meet a professor who sets you up with a nice new set of Hero Gear that's just what you need to save the world. But just as soon as you start to get your bearings, you'll have it all stripped away just as easily as you got it. After falling through the mysterious crater under Splatsville, you'll be tasked with traversing Alterna in search of the professor - as well as all your lost gear. Functionally, this means finding materials to unlock your old abilities in a skill tree.
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Much like in any skill tree, the ultimate goal with your Hero Gear is to work down each available track to get to the upgrades you want the most. In this case, you should be able to unlock the entire thing if you're searching thoroughly for materials in Alterna. There's an additional twist to this tree, though. Unlike many other skill trees, which allow you to upgrade freely along a certain path, the skills here are locked in small groups that you need to unlock using a separate material: Sardinium.
The skills are listed in their locked groups (if they have them) in order from left to right and top to bottom.