For those just starting out with Terraria, the progression is a little ill defined.
1) Beat Eye of Cthulu. Either it is a random spawn on the surface at night, or you can summon it with the suspicious looking eye item. If the battle takes too long the eye will desummon, so if you're using the item summon it at the very beginning of the night. You can beat this with wooden armor and a bow, but copper/tin armor is recommended.
-Beat a Goblin Invasion-
2) Beat either the Brain of Cthulu, or the Eater of Worlds. The summoning items (Bloody Spine or Worm Food respectively) can be obtained via whatever item the creatures drop in your world, which depends upon your world gen. Gold/Platinum armor is recommended for this, though beating them with copper and an excellent ranged/penetrating weapon is a possibility.
3) Defeat the King Slime. Its spawn is assured whenever the slime rain falls from the sky. Nothing special here, but do note damage shrinks it while spawning more blue slimes. You get the ninja armor from the King Slime. Not great, but plenty of gel.
4) Defeat the queen bee. As long as you've gotten the area cleaned up, and have a high jump, this should be a breeze. Bee items are less than stellar, but decent weapons for a starting magic user.
5) It's Skeletron time. Make sure to have crimtane/demonite armor and a strong ranged weapon. A pair of wings, or at least the rocket boots from a goblin invasion would be an ideal way to make you mobile enough for this battle. Once you've put Skeletron down, head into the dungeon he protected, and loot some awesome items you'll be needing soon enough.
6) Now that you've defeated the three original bosses, it's time for the Wall of Flesh. By this point you've gotten to the hottest place in the world, and those wings have proven their value as a mining spree can easily open into a dangerous canyon. The wall of flesh requires either a good deal of time mining out a huge path in the underworld, or enough mobility to run and gun. Either way, Hellstone armor and weapons are a must. Once the Wall of Flesh is defeated, prepare for your world to change. Welcome the hard mode. Make sure to use the hammer you just got from the Wall of Flesh on at least 3 demon altars, so your world can be blessed with newer, and stronger, minerals. You'll need these minerals to fight off the next few bosses.
-Beat the Snowman Invasion (Late December)-
-Beat the Pirate Invasion-
-Beat the Goblin Invasion again, to get new stuff-
7) Now that you've seen the hallowed and your corruption/crimson has expanded, it's time to take on the first three bosses again. Only this time, they've been mechanized. Eye of Cthulu becomes The Twins, Eater of Worlds becomes The Destroyer, and Skeletron becomes Skeletron prime. These new bosses offer you hallowed bars and souls. You'll need the bars for armor and equipment, and the souls will give you new weapons and gear. Welcome to the first time that one set of armor has a variant for each main type of play (summoning is excluded, but the class really doesn't come into its own for a while.
8) Seven bosses down, and the next resides in the Jungle. Plantera is in the jungle, and while you're searching for its bulb you should definitely pick up as much Chlorophyte ore as possible. If you can manage it, craft Chlorophyte aremor to your tastes, and put Plantera down. If you're lucky, The Axe will await you when Plantera falls (an immensely powerful hammer+axe). No matter what, the key to the temple is dropped.
9) Finding the Temple is difficult, but defeating it proves even more challenging without either Specter armor, Shroomite armor, or Turtle armor. Hopefully you've gotten plenty of Chlorophyte, because it's the basis of all three of those armor types. Once the Golem goes down, you've got access to beetle armor and the Picksaw (pickaxe+axe).
-Defeat the Pumpkin Moon event, or at least survive long enough to get the Pumpkin Sword-
-Defeat the Martian Madness. It will probably destroy you without a constructed arena-
10) It's time to go fishing. Unfortunately, that means it's time for Duke Fishron. Once it's put to bed you'll have access to the Fishron wings and the Flairon. Welcome to a dream for Melee characters, ranged and homing melee bubbles.
-Defeat the Frost Moon event. It's just an upgraded Pumpkin Moon. The great prizes here are for magic users.-
11) Begin the Lunar Event. You will get your crap wrecked. Once you manage to defeat three of the 4 towers go craft some new weapons. Each new fragment gives you access to more amazing stuff. Magic users should cap the Stardust, Vortex will fall to projectile weapons, Solar is for melee players, and the Nebula tower rewards magic users. Once you've got new weapons, prepare to get wrecked. Kill the last tower, and 60 seconds later the Moon Lord arrives. Good luck with that. It's tough as nails, but will reward you with the absolute best weapons in the game. Combine that with Luminite, and you can craft the best armor. Welcome to the end of Terraria. Murder enough Moor Lords, and you can get some of the very best stuff the game has to offer. Welcome to Meowmere.
That seems so very short of a description, but Terraria is about the discovery. Between that, and building a lava moat, it's a game worth experiencing.