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;
v
e
r
t
i
c
a
l
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
floating islands lucky horseshoe and
shiny red balloon
Build a skybridge, craft meteor shot, 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: