How do I survive in the Underworld?

You really should have at least an Obsidian Skull before you attempt to tackle the Underworld. It makes you immune to the damage from Hellstone and Meteorite Ore and simply requires 20 Obsidian to craft at a Furnace. However, I strongly recommend that you craft yourself an Obsidian Shield as this also provides immunity to knockback effects, which can be incredibly dangerous when you're mostly surrounded by lava! For that you'd need to head to the Dungeon (and kill Skeletron if you haven't already) and open the chests there until you find a Cobalt Shield, then combine it with the Obsidian Skull at the Tinkerer's Workshop

An appropriate set of armour would be extremely beneficial as well (you didn't list what armour you're currently using).

If you're more of a melee player at heart, then you should consider getting either the Shadow Armour or the Crimson Armour depending upon whether your world has Corruption or Crimson.

If you're more of a ranged player, then consider getting a set of Meteor Armour and a Space Gun to go with it. If no meteors have landed in your world yet, head to the Corruption/Crimson and smash a Shadow Orb or Crimson Heart or two (do it at night for a 50% chance of a meteor either immediately or the next night, depending on the time).

Next up, make sure you've got some mobility-enhancing items with you as well. Some form of Grappling Hook is extremely important as you won't be able to make a set of Wings yet. The hook will allow you to cross over the lava lakes much more easily, and can save your life in a pinch.

Make sure you have a stack or two of healing potions as well. Obsidian Skin Potions are also an extremely useful tool for mining Hellstone and generally improving survivability due to the lava damage immunity it grants, however you need Fireblossom as one of the ingredients so you probably won't be able to make any until you've played around in the Underworld for a bit first (you can sometimes find them in pots in the Underworld, so smash any that you find).

Also remember that an Imp's Burning Sphere projectile can be destroyed by simply swinging your sword at it when it gets close - completely negating any damage you might have taken.

First thing before you go to the Underworld: Get a Obsidian Skull. Find water next to a pool of lava and mine it so the two collide and make Obsidian, mine that and make a Obsidian Skull. It grants immunity to Hellstone blocks. Ranged weapons are a must, a Demonite or Crimson Bow could do the trick, or maybe a gun from a Shadow Orb / Crimson Orb. Stock on ammo for those, take some health potions, pick the best hookshot you have and use the ceiling of the Underworld to get around lava pools. To mine Hellstone, enter the buildings and mine them through the wall, so the lava won't flood in and kill you. If you're more desperate, start making Obsidian Potions for lava immunity. Also, if you have enough resources, make bridges across lava pools so you don't get knocked back and fall into lava and lose your gold.

Just get an obsidian skull/sheild and an obsidian skin potion and you should do okay

You might want to keep a peace candle around with you if monsters are too much of a priority to fight. I would suggest having a set of meteorite armour on, along with items that allow double jumping, increased speed and (smooth) flight (i.e. Flying Carpet, Cloud in a Bottle, Bundle of Balloons, Rocket Boots, Sailfish Boots, and any horseshoe balloon).

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