There's ways to metagame and leapfrog ahead by huge amounts in the Terraria tier system, but if you want to take a more relaxed approach, while still being efficient, this is about how I usually do it.
To start, I'll make a wooden sword since I hate the stabbing weapons such as the starter sword. I next chop a bunch of wood and then begin working on my "hellevator" down to the magma level. While digging my hellevator I'll usually make a silver or iron pick to speed it up some. Once I finally reach the magma level, I've usually naturally come across enough gold to make at the least a golden bow, which I will craft to take down the eye boss. Any leftover gold, silver, or iron from my hellevator project I will use to make helm / chest / legs. I then build a very basic arena to fight the eye, and farm enough lenses to spawn him. Taking out the eye with the golden bow + fire arrows is fairly easy, and this gives you demonite ore.
One eye kill will give you enough ore to make the Light's Bane sword, which is all you need to take out the eater of worlds. I will next buy about 5-7 dynamite, gather up some consumables, and go off to find the corruption. Once found, I'll use dynamite to blast through to some orbs and spawn the eater of worlds. You may want to use some wooden platforms in the area of the third orb before spawning him to have an easier time with the kill, but he isn't very hard if you use potion buffs and the Light's Bane sword. Killing him gets you shadow scales (and more demonite). First thing to do is make a Nightmare Pickaxe, With the remaining ore/scales, start working on the shadow armor suit. You want the full 3 pieces of the suit for the set bonus, so farm up the items to spawn eater or eye again until you have enough materials to finish the suit.
Next order of business is to get some of the really nice accessories from the dungeon, but in order to do that you first need to defeat Skeletron. I like the minishark to kill both Skeleton and Wall of Flesh (the next boss), so I'll scrounge up the money to buy a minishark and a ton of ammunition to kill him. Again, make a basic arena outside the dungeon to dodge him easier, then take him out.
From the dungeon, you're going to want to get a cobalt shield, a shadow key, a few extra golden keys, and optionally a muramasa. You will also want to spend some time in the jungle to craft an ivy whip, and optionally find a blade of grass. Next, use some gravity potions and find your floating islands. Using the golden keys you found, try to get a golden horseshoe and a shiny red baloon. Finally, upgrade your cobalt shield and golden horseshoe with obsidian skulls to make the higher tier accessories.
Use any extra cash you have with your tinker (goblins should have invaded by now, which lets you find him soon after) to get the warding property on as many accessories as you have. Having 5 accessories "of warding" is an extreme defensive boost. If it's not an end-game accessory, then lower defense mods like armored or guarding are fine.
At this point you have a choice. You can skip the hellstone gear, or spend time obtaining that. If you decide to do the hellstone tier, then mine that and make a full suit. At the bare minimum, you must make the molten pickaxe, as it's required anyway before the next tier. Crafting a fiery greatsword, then combining it with your old light's bane sword, the blade of grass, and muramasa gets you the best pre-hardmode sword (though I prefer minishark for wall of flesh anyway). One final step I'd recommend before spawning hardmode is go out and gather 20 meteorite bars + 10 of the same kind of gem (any) to make a phaseblade. Phaseblades are kind of meh, but their hardmode upgrade is excellent.
Now setup a long bridge to fight wall of flesh, spawn him, and unlock hard mode. Optionally, keep spawning and farming wall of flesh until you get the emblem for your favored weapon type, as it's an endgame accessory.
You are now in hard mode. The end-game ultimate gear is hallowed, and it requires you to craft 1 of every single hardmode tier, so at this point there's no more skipping around, you have to make it all. The one thing I would recommend you try to do as soon as possible is mine 50 crystal shards. Combine those shards with your phaseblade, which will upgrade it to a phase sabre. Phase sabres are $$, they have high attack power, fast attack speed, large reach, and are auto-fire. I'd recommend you spend some gold to get the legendary modifier on the sabre with the tinker if you can.
As to the hard mode bosses order of progression, once you get a phase sabre, a pair of wings, some of the hardmode armor, and most/all of your accessories reforged to "of warding", the destroyer is a fairly easy kill. Farming The Destroyer gets you souls of might, which let you get the final tier of breastplate, sword, and most importantly, the megashark. The megashark weapon is almost brokenly overpowered in Terraria, though it eats through ammunition like woah. Using the megashark and a proper arena to fight in, you can take out the other 2 hard mode bosses without much issue.
Edit: looks like Sinzar already made a nice little guide.
The only thing i would made a point to recommend is flails are way way way awesome. Ball o' Hurt and Sunfury and then later on the Dao of Pow are all excellent, especially against the Eater of Worlds and The Destroyer. This is made more important because The Destroyer is the gateway to the Megashark, which is typically the weapon of choice for boss killin'.
With the Dao of Pow you can grapple onto a wall (one made of molten bricks without an obsidian skull for some sweet sweet damage immunity due to taking minor damage from the brick), hold down the attack button, and watch the mecha worm Destroyer's HP drop by the thousands. Can summon several of them in a night. You just have to be careful to know where the Destroyer's head is when it dies, otherwise you could end up never finding its' drops
That's a lot more content that I originally played. I might fire it up again and restart the game.
If you've been away from Terraria for a while, but want to do a play through again, I'd recommend waiting a few more weeks. Patch 1.2 is due out soonish, and it's a fairly substantial content patch, with multiple new bosses, new tiers of gear, new enemies, items, etc etc etc. Basically a mini-expansion. To my knowledge there is no official release date for the patch aside from late june / early july.
Would anyone recommend some beginner mods I could add to my server? There are 2-3 of us playing. UI, hotkeys, transportation/travel mods are top of my list I think. Content is also always nice.
And somewhat related - is there any item I could craft that eases traveling? I know about the grappling hook and the mirror. Is there anything that lets you teleport to other points in the world - other than to spawn?
For me the worst part of this game by far is that you can't explore underground, log out for the night, come back and start where you left off. I need 2 teleport points. I have spawn with the mirror (or will eventually). Any mod let's me set one extra teleport point? I don't care if it requires building extra stuff like teleport pads or what not. I'd welcome it actually. I have to have something that lets me teleport to a customizable point in the world.
The mod Adventure of Red Cloud adds in 2 more mirrors. They let you set waypoints, then teleport to them at will. The mod is basically an rpg'ish adventure map, but it has a standalone free build version. The mod adds a huge amount of other things as well, including gameplay overhauls, so it may be overkill to use the mod just for the waypoint mirrrors.
Modding in Terraria is very simple. A few minutes reading the mod wiki should be all you need to know how to make / edit mods. If you wanted, you could just rip off his 2 mirror items and make your own custom mod that only adds the mirrors. It's just text editing a file or two to make it.
Speaking from over a year ago memory and I'm sure many times removed, so probably wrong silver -> gold -> shadow -> meteor = jungle -> hell -> endgame/dungeon. You used to be able to skip the meteor and jungle tier fairly easily once you got the hang of playing around the lava. A sky island gear is obtainable around the same time as meteor/jungle.
edit: oops, didn't realize I was replying to the last post of the 3rd page.
Holy inventory management... Any mod out there that improves it? The quick stack/loot all/deposit all are nice but just don't cut it. I need a "sort" button on my inventory and chests to put weapons together, armor together, potions together etc... for one thing.