I typically focus first on things that give me more iridium: pickaxe (faster mining), crystalariums (jade = staircases and ruby = spicy eels), iridium band (big upgrade over either magnet or glowstone ring).
Then I upgrade the other important tools (axe, hoe, water can) and get iridium sprinklers, depending on which is more pressing in a certain playthrough. I only upgrade stuff like the pan or the trash can when I don’t know what to do with iridium any more.
It does but it’s unreliable; mining is faster once you have it set up properly.