How to Find Good Loot Early in Terraria

Water Chests

Be smart in water. You'll need okay armor (preferably cactus) and, if you're not as experienced, have an item which will help you out (cheapest is wood platforms).

Travel to the west/east border of the world. You should find an ocean. (Be careful of the Corruption/Crimson!)

Travel underwater. Once you're down there, make a mad dash! just select "Loot All" for all the chests you can find then rush back to surface. Now you can take a look at all your goodies!

Skyware Chests

Find a Gravitation Potion. Be sure to bring many, as it wears out in 3 minutes! You can use a Gravity Globe, but what's the point of that if you've already beaten the Moon Lord?

Use the Gravitation Potion. Press Up on your keyboard and when you reach space, rapidly tap Up.

When you find a floating island, land on it. Try to land lightly, or you know what's going to happen!

Open the door and pick through the contents. Shut the door, or else the harpies will shoot you!

Ivy Chests

Find a Jungle Biome. This is indicated by lime green grass, dark brown dirt, mahogany trees, and those pesky Jungle Slimes.

Look around the surface. If you're lucky enough, there will be some ivy chests in areas shaded by blocks.

Find an opening in the ground. If you're not that desperate skip the upcoming parts of this section.

Venture into the Underground Jungle. You can find a Lihzard Temple, providing even more loot, and you don't even need the Temple Key!

Gold Chests (Mining Shacks)

Go underground.

Find chests full of loot in wooden buildings. These always contain something useful such as a flare gun or accessory.

Gold Chests (Dungeon)

Defeat Skeletron. You need him to access the Dungeon.

Kill enemies. Most preferably the Dungeon Slime, as it drops a key every time.

Find Gold Chests and use the keys. If you're lucky enough, you might find 2 keys in a chest, and even a Shadow Key!

Shadow Chest

Find the aforementioned Shadow Key. It's in the Dungeon, from chests.

Venture into the Underworld. I greatly recommend Shadow/Crimson armor for melee users, Jungle Armor for magic users, and Necro Armor for ranged.

Look for Ruined Houses. They contain the chest AND pots, giving extra loot.

Once you have enough, swiftly retreat. The Underworld is a very dangerous place, and if you stay for much longer, you'll probably get killed...