Terraria – a reliable method to locate floating islands 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.

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.