Items that are desirable to get, roughly in order, when starting a new game:

item benefit important things to do to obtain it
iron anvil can't craft metal otherwise 50 silver coins, or 5 iron bars
mining helmet (PC version) too dark otherwise 8 gold, get the Merchant to move in
vilethorn powerful against early mobs farm orbs in Corruption
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climbing claws avoids falling dmg,
very limited upward movement
search surface chests
ivy whip easier vertical travel reach the cavern layer, farm skeletons, get a hook
rocket boots easier vertical travel
allows hellevator
summon and beat the Goblin Army, find and rescue the Goblin Tinkerer
wings enter Hardmode
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Hermes Boots,
Flurry Boots,
or Sailfish Boots
faster speed
mount faster speed
asphalt faster speed enter Hardmode, defeat one Mechanical Boss, and go talk to the Steampunker
floating islands lucky horseshoe and
shiny red balloon
Build a skybridge, craft meteor shot, become a ghost... many ways.
cobalt shield knockback immunity beat Skeletron, farm golden chests in the dungeon
obsidian skull fire immunity
(but not from lava)
gather 20 obsidian


pickaxe progression: gold pickaxe ⇒ nightmare pickaxe


Less essential things: