PC Update: You can actually get one of (if not) the best pre-hardmode pick in the game by fishing in the ocean. You will need roughly 50% fishing power from bait and fishing pole combined (Enchanted Nightcrawlers and the Reinforced Fishing Pole will suffice), but the Reaver Shark will also suffice in mining up everything including the initial hard mode ore.
Original Post: You are skipping a step. You need to make a Nightmare or Deathbringer pickaxe. You get the ore and other components from defeating either the Eater of Worlds or Brain of Cthulhu respectively. These axes can then mine the Hellstone and you can keep moving forward.
There are two ways to summon these bosses, but they parallel each other. You can craft the appropriate summoning item at an Altar but more likely you will want to destroy the Crimson Hearts or Corruption Orbs. They are located at the bottom of the pits that naturally form in both of these zones. You will likely not be able to break the rocks but you can use TNT and such to destroy the rocks to get access to them. Every three of these items you break will summon a boss. You can usually summon 3-4 in this manner. After that you will need the items.
Both bosses 'hover' around. The brain literally does it appearing and disappearing to then swoop at you. The Eater of Worlds is a giant worm that will circle around you and try to hit you. The general strategies for both of them are to build an arena where you can jump up and down between 2-3 levels to help with placement. You may want to include some Camp Fires or healing lamps or the like to help your passive regeneration of health. Ranged weapons will be handy for the eye ball but honestly for both of these bosses you want multi-penetration items like spears that will hit continuously when they make contact.
For a list of Pickaxes and what they are made from and what they can mind, check out the Terraria Wiki. For recipes for the items you can check out the Worm Food and the Bloody Spine to summon the bosses.