How do I progress past the initial tier of gear/crafting in Terraria?

You can go exploring some corrupted/crimson areas (randomly picked at world generation) to destroy the shadow orbs/crimson heart waiting for you at the bottom of chasms.

Each shadow orb/crimson heart destroyed will provide you some loot, and make a meteorite land on the world. Once the meteorite has fallen, go find the site, and mine the meteorite ore. This will allow you to craft some meteorite gear and weapons.

Once you've destroyed three shadow orbs/crimson heart, it will make the Eater of Worlds/Brain of Cthulhu spawn, allowing you to fight two different bosses.

Note: The shadow orbs are protected behind some ebonstone/crimstone, which you will not be able to mine with your current pickaxe. Just buy some dynamite from the Demolitionist and use it to access the shadow orb. Or you can buy purification powder from the Dryad, which turns the ebonstone/crimstone into the easily destroyable standard stone.

You can take on the Eye of Cthulu with a bow and a pile of potions. Farm lenses from the Demon Eyes at night. With 6 lenses at an Altar, you can craft a Suspicious Looking Eye to summon the Eye of Cthulhu. Be sure to use the eye at night or the summon won't work.

Once defeated, use the demonite/crimtane ore dropped to craft a Nightmare/Deathbringer Pickaxe which you can use to get hellstone and obsidian to make hellstone bars. Hellstone bars are used to make the highest pre-hardmode gear.

If you can get past the Eye Of Cthulhu, the dryad will show up. She will sell you purification dust which will let you convert ebonstone to regular stone and allow you to get to those tasty shadow orbs.