Terraria – How to progress past the initial tier of gear/crafting in Terraria

Here is what I ended up doing:

• Searched for Gold to create gold armor and weapons (doubtful you'll get full gold gear)
• Killed the Eye several times for Demonite Ore
• Destroyed two Shadow Orbs for their drops (the third spawns the boss Eater of Worlds)
• Dynamite the Meteorite and Magic Mirror out of there (after enough of it is mined out, the meteor heads will stop spawning) then use a gold Pickaxe or higher to mine the Meteorite.
• Floating Islands. Craft your skyways during the day, the spawn rates are higher at night for Harpys. Remember these skyways disrupt Meteors, making them very non-profitable.
• Explore under the surface! There are plenty of hearts and chests filled with great magic items! You don't need to go too deep either. Try natural caves.

Now is also an excellent time to farm hearts and stars, and participate in the Blood Moon event for cash. I think the game gets really fun once you get over this little hump in character progressions.

I tended to make little goals like:

1. I will mine this meteor.
2. Then I will look for hearts.
3. Then I will search for floating islands.
4. Then I will destroy another shadow orb.

You get the idea.

EDIT: Sorry I can't provide links, at my desk and unable to reach the wiki.

EDIT2: See this question/answer as well for a similar problem.

The enchanted boomerang can be difficult to use against the various Eaters that will swarm you in the Corruption. It's not quite fast enough to take them all out before they overwhelm you or knock you back into one of the many caverns. They like to surround and swarm you very closely.

For traversing the Corruption quickly, I'd recommend:

• The Shiny Red Balloon, Cloud in a Bottle, and/or the Rocket Boots to help you handle jumping over the caverns and over the hills. Mobility is key if you want to outrun the Eaters.
• A sword or other melee weapon to quickly break through the thorn-type plants that hurt you and knock you back and to take care of the Eaters. The Big Eaters have 45 max health, so having one that does enough damage to kill them in 2-3 hits is helpful. Breaking down the thorns will also increase the speed at which you can climb the hills.
• Ranged weapons with a high recast time such as bows and arrows, flails, or spells like the Vilethorn can be handy in trying to keep the Eaters away from you, but I've found that if you stick around too long in one place, you will get swarmed anyway and need to use the melee weapons for a quicker takedown.
• A Grappling Hook or Ivy Whip in case you fall. If you don't have a Lucky Horseshoe, it's very likely you will splat, as the caverns are usually very deep in the Corruption.
• When I first crossed, I kept a bunch of Wood Platforms on me so I could build them over the caverns. This makes re-crossing later on much easier and faster, as you can just run over the caverns, and you won't get knocked into them by the Eaters.

Tunneling under the corruption would be incredibly time consuming, and I'm not sure how far the Ebonstone goes down in your world, so you may not be able to quickly do it at all. Tunneling through Ebonstone would require either a Nightmare/Molten Pickaxe, or a Dryad to sell you Purification Powder so you could mine through it with lesser pickaxes. A skybridge would be less time consuming and is also a viable option, but keep in mind that if you have a Meteorite fall later on, if it hits a skybridge, your crash site will end up with only a few blocks of Meteorite. :(

If you can't handle the Corruption right now, the Dungeon and Skeletron are probably a bit too difficult at the moment with your current gear. If you haven't yet, focus on taking down the Eye of Cthulu and Eater of Worlds so you can make Demonite gear, which will prepare you better for the dungeon. Remember that the Eater of Worlds' individual segments also drop Shadow Scales, so even if you fail in completely killing him, you should be able to get at least a few drops.