To get extra sockets you'll need artificer's orbs which work pretty straightforwardly - just right-click the orb, then left-click your target piece. The timing on using these is pretty crucial since socket limits are pretty strict - chests and two-handers can fit 2 sockets max while one-handers, shields, helmets, gloves and boots all cap at 1 socket. Really important to note that accessories can't take sockets at all so don't waste orbs trying.

For getting the orbs themselves, you've got two reliable paths - either find them as drops while playing, or break down socketed gear at the salvage bench which gives you artificer's shards. Once you collect 10 shards, you can combine them into a fresh orb. The salvaging process is actually pretty consistent, just make sure you're breaking down appropriate gear.

Late-game gets interesting with vaal orbs, which can push you past normal socket limits by corrupting gear - potentially getting a third socket on two-handers or chests. The catch is that corruption locks your item from further changes, so the timing on when to use these is super critical. Also worth mentioning that once you slot runes or soul cores, that decision is permanent - the positioning and timing of socket usage becomes really important especially on your martial weapons or endgame armor where mistakes are extra costly.