What should i do next?

The next step is nightmare pickaxe. Beat Eater of World and Eye of Cthulhu and you will get the ore needed to make one.

Yes there is a dungeon in every world, the third boss is Skeletron and stands outside. Talk to him at night before going inside. There are lots of good items in the dungeon. That is where most of the best gear is found.

read a FAQ?

With Skeletron, you have to beat him during the night, and same for Eye of Cthulhu, but the Eater of Worlds can be beaten at anytime. I suggest beating Eye of Cthulhu first, since it will make the Dryad appear and then you can use purification powder to dig into the corruption (cheaper than using dynamite) since the powder turns the ebestone (only last 2 pickaxes can mine it, which are nightmare and molten) into normal stone. Then smash 3 orbs and beat Eater of Worlds. Make the nightmare pickaxe, and have fun expanding HP/MP, crafting better weapons, etc...

On a note, the nightmare pickaxe can dig up anything.

For the nightmare pickaxe... Get 10 lens (you get them for slaying demon eyes), go to a corruption and look for demon altars inside the chasms(demon alters look like creepy looking mouths that are grey). Don't try to destroy it with a hammer, you'll kill yourself.Those things work like a workbench.On the demon alter you can craft a suspicious looking eye, which you can summon by left clicking. Summon it at night and defeat it. I'm too lazy to write down the strategy, but you can see it at the faq section. Next step is to kill soul eaters and get 30 rotten chunks, and go to the corruption look for vile mushrooms, and turn them to vile powder on an alchemy station I think.Create 30 of those. Each vile mushroom creates 5 powder.Go to the demon alter to make worm bait, that you can also summon. Summon it at corruption and kill it. I also won't provide a strategy here. Once you defeated both of them, you should get a bunch of demonite ore and shadow scales. Turn demonite ore into demonite bars and with dome demonite and some shadow scales you can get armor and weapons, including the nightmare pickaxe. There ya go!

Eater of Worlds drops the materials you need.

You can only gain Demonite Ore from bosses like Eye of Cthulu (which is the "easiest" so far - discounting the rare King Slime of which there is no method to summon it yet).

Despite what the others are saying here, killing a boss, even with Gold-type equipment, is easier said than done. Your best bet is the Eye of Cthulu first, which you can summon by gathering 6 lenses (at this time of writing) and merging them into an item that can summon it at a Demon Altar (which you have to find underground too).

You really, really, will not want to fight Eater of Worlds and Skeletron with Gold-type equipment, despite the suggestions of the others here.

Fighting a boss is another question entirely, so I am not going to write in detail about this here. Suffice to say that you are going to need to be very mobile when fighting it, and I recommend building several tiers of wooden platforms that you can hop up and down while fighting a boss. You ought to keep in mind too that fighting a boss is a battle of attrition.

To mine corruption, you need a Nightmare pickaxe. To get that you need some Demonite ore (which you can get quite easily from the Eye of Cthulu, and you summon him by combining 6 lenses at a demon alter and using the result at night, which is called the "suspicious looking-eye") and some shadow scales, which you get by defeating a segment of the Eater of Worlds, which can be summoned two ways: 1: break three black orbs, which you find in chasms in the corruption. To get at them, however, you need purification powder, which the Dryad sells for 75 copper each. The dryad comes after you defeat a boss monster. 2: summon it with worm food, which you make with rotten meat (soul eaters drop them) and worm teeth (devourers drops them), then you combine them at a demon alter. You can only summon him in the corruption and at night. With the Nightmare pickaxe, you can mine obsidian and hellstone, and mine stone and other materials more quickly. To make hellstone bars, you have to have 2 obsidian and 6 hellstone ore (which you get in the underworld) and you have to have a hellforge to make the bars. I hope this helps!

To get the Nightmare Pickaxe, you must kill the Eye of Cthulhu and the Eater of Worlds.

The Eye of Cthulhu is summoned by crafting 10 lenses together at a Demon Altar to make a Suspicious Looking Eye, which is used at night.
The Eater of Worlds is summoned by breaking 3 Shadow Orbs or Crafting 30 Vile Powder and 15 Rotten Chunks together to make Worm Food, which is used in Corruption.

The Eye of Cthulhu drops Demonite ore upon death and the Eater of Worlds drops Shadow Scales as well as Demonite ore upon death.