Robb's answer is very good, but I am going to offer up a more detailed, and alternate path.

The high level version is:

1. Beeline for shadow stuff, get into the dungeon, loot the skyworld.
2. Tunnel to the bottom of the world so we can mine Hellstone and kill the Wall of Flesh (And then farm him for fun things).
3. Hardmode!

The Early Game, or How I Farmed the Eater of Worlds for fun and profit

Our goal here is the famed Eater. He drops shadow ore and shadow scale, which we can use to make gear that will tide us over until we can farm Hellstone at the bottom of the world. Honestly, it's probably easier to get a full set of shadow armor than it is to get a full set of gold, depending on how lucky you are with ore field spawns in the natural caves. We want meteor ore, but it's a luxury, and we're not going to maximize meteor spawns (which would mean breaking one shadow orb a day only).

Starting out: Explore the natural caves that generate in the world, looking for chests, hearts, and ore (iron, silver, or gold only). Note there are a couple of things in particular we want from chests. Wood chests can contain iron bars and Gold chests can contain silver and gold bars. These are exceptional finds; a stack of 10 gold bars can take awhile to farm naturally. We also need bombs, not grenades. Dynamite is great but it's rarer; we need as many bombs and dynamite as we can get our hands on. Also, we need spiderwebs to make a bed (takes 50 webs; eventually we will want more than one bed, most likely). Collect gems too; we need at least 10 of one color.

For our home, it's best to set up shop near a swath corruption or the dungeon. Ideally, nearish to both.

A magic mirror is borderline critical. It will really help us in these next parts, but remember you can save and quit to simulate "town portal". Boots of speed and the Cloud Bottle are the only two accessories we really have a hearts set on. An enchanted boomerang is a minor luxury. Horde money, of course.

Crafting: Iron pickaxe first, then make armor from the best stuff we have. Make Iron broadsword and bow after this, if desired. Make 3 iron chains and save them for the grapple hook. Hope to farm the hook down in the stone layer (it makes a lot of things easier but is not strictly speaking necessary for the next part).

Savvy look ahead type stuff farm death weed, glowing mushrooms, cacti, daybloom, mushrooms. Have some extra iron ore lying around. These are ingredients to make some key potions.

The Corruption

Needs: 7-8 hearts, minimum (140-160 hp, but if you're very good you can get by with less), at least iron armor. Lots and lots of bombs/dynamite. A grappling hook or some other way to survive the falls (we'd prefer to speed this part up, but you could use a series of platforms/careful hugging of walls when they go concave/etc to get to the bottom). And a little bit of luck. Ideally, we could bring regular healing potions, thorns potions, iron skin potions, and regeneration potions.

Ordinary play would see us break one shadow orb a day, to maximize the chance for meteors to fall. Meteor ore has a few awesome and a few solid uses. But it's not critical to our strategy. With iron+ armor and an iron skin potion we're now quite resilient. A band of regeneration and a regen potion will probably see us not needing to heal too much. Our biggest problem is going to be hurting the Eater himself. A little luck will solve that problem shortly.

We want to break 2 orbs ASAP. Our hope? A Ball O' Hurt drops. The Ball of hurt, + the above setup, will make the Eater fight almost too easy, as the ball will frequently hit 3-8 sections at a time, doing multiple hits to each. Patience, and the occasional heal, and he's ours. And then we can easily kill him 3-5 times, and make a hole bunch of awesome things. What if we don't get the ball (20% drop chance from an orb)? Well, you have a few options. Note: you are more likely to segment the Eater a lot with the Ball. With Gold armor and those potions, I have not found this to be an issue. YMMV.

Ranged weapons bows are slower than we like, but with unholy arrows you get piercing (need worm teeth to make these). A musket is ok, better with meteor shot, though that requires meteor farming. A better option? Shurikens. Save all Shurikens while you are doing your initial exploring. Take a big ole stack into the fight. They pierce, which again means "worm splitting" but you should be well defended enough not to care. I finally tried this method and they worked wonderfully. It's much easier to have 200+ shurikens lying around than hope for a Ball O Hurt as well. IIRC, I used about 130 on my shuriken kill.

Vilethorn is excellent, but it requires a lot of mana to use. Note, even if you don't have the mana for more than a swing or two, keep it on switch (unless you have Hurt), and switch to it every time mana tops off.

Desperation: if you have to fight the goblin army, you will collect a lot of spiked balls. Line the bottom of your "arena" with them, and the Eater will take a bunch of damage over time coming and going.

Also, you can always crack a third orb and just let the eater kill you. There's more than enough orbs on a medium sized world to farm him many times over. A Ball O' Hurt is really our heart's desire.

The actual fight isn't too hard- make an arena with wooden planks on several levels in one of the more open areas in the huge horizontal layer in this section of the corruption. Wall off most of the top and part of the sides, to minimize unwanted guests getting in. Once you open the third orb, head straight to it.

What to craft: Shadow pickaxe first. Then shadow armor (will need scales here). Then a melee weapon and maybe a bow, depending on your needs. If you've managed a phase blade, don't worry about making a melee weapon.

Meteor ore: we want this eventually, but our only needs here are really meteor shot and enough bars to make a phase blade perhaps (20 meteor bars, 10 gems of a single color). Using dynamite to mine it is a huge time saver and also a frustration saver. We don't care about the armor. A hamaxe is a luxury. The Space gun - made from meteor among other things - is nice but also mana intensive (less so than Vilethorn). Note, large weapons with good attack speeds can be very effective against the Eater.

Welcome to the Jungle

You don't need to go to the jungle at this point (or before now, by the by that's an option but this path is probably easier for neophytes), but you can. The Jungle has one thing we covet above all others: the ability to make the Ivy Whip, which is a huge upgrade to the grapple hook. That said, the Grass Sword and the Grass Chakram can be very useful, so if you want to farm them, do so. But you will want an Ivy Whip eventually.

The Dungeon: A Golden Ticket to Golden Keys

Once you have a grapple hook (or the whip), shadow gear, a decent ranged option, and a stack of potions, it's time to kill skeletor. Once again you want to make an arena outside the dungeon. A musket is great for this fight, don't necessarily waste meteor ammo (we have something else we want to use that for). But the Ball or some other melee weapon can be used. So can a Grass Chakram. Wait until night, summon Skeletron forth, and then killing.

Needs: there are a lot of nice items in the dungeons, but we have only 4 needs really.

Free the NPC that will spawn inside: make sure you have an empty room to hold them.

Find a shadow key in a chest: just because.

Find a cobalt shield in a chest: we'll want it later.

Farm golden keys: Ww need them to open the vast majority of chests in the dungeon. But we also need them to loot the sky islands, if we want to go that route (I recommend it).

There are other nice items, don't get me wrong. Muramasa, some nice magic options. An upgrade to the Ball O' Hurt. And you will make a lot of money. Also, if you aren't topped off in heart containers you will be after a couple of dungeon runs. Also, the handgun has crafting potential.

Sky Islands in The Jetstream

Again, not necessary, but the path to real ultimate power is eased with a few toys found therein: The Golden Horsehoe (no more damage from falling!) and red balloon (jump higher). It does take some effort to get up to them (and to find them, in fact, unless you use a mapping program for that).

The House At The Bottom of The World

And now, we're ready. We have a shadow pickaxe. We have shadow armor. We have many wonderful toys. It's time to dig the so called "hellevator", straight down to the bottom. This will take awhile, but it can be an oddly meditative experience I've found. Put on your favorite music (I used to use Big Dipper's Anthology), kick back and relax as you shape the world into something of your own making.

First order order of business? Farm Hellstone. It's time consuming, but we want a full set of Hellstone gear and a Hellstone pickaxe. All else is optional. If you can find a shadow chest, loot it. There's good stuff in those.

What next? At this point you really need to hit the jungle if you haven't. Now is also a good time to hit the sky islands, because we're about to move this game into some Nth Gear that is both exciting and difficult to handle at the same time.

Lastly, by this point you should have freed several NPCs, including the Goblin Tinker. Now is a good time to avail yourself of his excellent services. You can make several "super accessories" and should so so. Once you are very wealthy, you can reforge items to get better bonuses on them (very expensive, though). He does sell rocket boots. Trust me, you want the rocket boots.

The easiest way to kill the Wall of Flesh is build a hugely long horizontal platform (yes, this will take awhile). At the far end, you summon him and then be about your business. The easiest way to kill him is ranged weapons by far. Meteor shot is invaluable because it will pierce, and you can hit multiple things. I recommend using the Minishark (purchased from the arms dealer). It has low base damage, but it shoots so fast and will conserve quite a bit of ammo. Killing the Wall of Flesh will activate Hard Mode. You can farm him over and over, getting some rather choice drops and making quite a bit of money. Advanced players would use a combination of feather fall, speed, and gravitation or maybe water walking, obsidian skin, and speed to avoid needing to make the bridge. But I like making the bridge. It's my time to reflect on all the fun that came before, and all the new friends I get to make after killing the Wall.

And Hardmode is now open and waiting for you to explore it. It's much more difficult once you're in the new content (as it were, and no it's not everywhere at once). However, it's incredibly rewarding. 3 New bosses, a bunch of new enemies, and a boatload of new items all waiting for you to play with. You will be feeling invincible once you're ready to face the Wall of Flesh and Hardmode will disabuse you of that notion. But you'll claw your way back to invincibility and then beyond, to double plus invincibility with a helping of Power Puff Girls for good measure.