How do I progress beyond obtaining Wood Armor?

Solution 1:

Downwards. You're looking for metals like lead/tin and copper/iron to start off with. (only one set will spawn in your world.) Sky fortresses are a bit hard to find, save it for later. Right now, your main goals are:

1. Get metal
2. Make decent gear
3. Look for more exotic materials and trinkets in varied biomes
4. Create boss summoners and destroy the resulting bosses
5. Dig downwards and see what's going on down there

If you can get your hands on a Magic Mirror or Recall Potions, your life will become a lot easier!

Solution 2:

Progression in Terraria is basically measured by boss kills, especially the game-changing Wall of Flesh, enabling the so called hard-mode.

The objective in the beginning of the game is to get strong enough to kill the first boss, the Eye of Cthulhu. The game itself has a "barrier" for confronting the Eye, requiring 200 life and 10 defense (for at least one player) and three NPCs living in your town. Note this will only spawn the Eye of Cthulhu one time, if it was not previously defeated in that world. There's another (repeatable) way to summon the Eye you can look for.

To be able to achieve this HP and Defense requirement, the best solution is to explore the Cavern layer, notably in the easier "Forest" area (not jungle, ice or crimsom/corruption areas, the normal dirt-and-stone ones). You know you're in the cavern layer when the background doesn't allow you to put torches on it. In the cavern layer you will find very useful artifacts and powerful armour/weapons, paired with Heart Crystal to increase your maximum life. You can also get some minerals for better tools and weapons if needed.

If you want to have an even easier fight against the Eye, use some Shurikens in the fight, its a great weapon against it. After the Eye is defeated you should start to look into the corruption or crimson area (whatever is generated in your world) for the second boss, keeping in mind the Eye drops Crimtane or Demonite ore, allowing for some much better gear/tools.

Note this pattern of "kill boss, enable new gear, get new gear, kill next boss" keeps going on until the game is over story-wise (there are still building, rare-artifact chasing, events like Helloween, etc).