Terraria is an interesting game in how the boundaries are "soft". The player is often not stopped by an insurmountable indestructible wall that disappears after some arbitrary deed in accomplished; their progression is stopped by difficult enemies, time-consuming mining or construction of bypassing architecture; this makes the player feel the need to go back to an easier area and gather up materials and equipment to ensure that they can survive the difficult new area. As the player travels the surface they will find deserts containing somewhat more powerful enemies than that of the starting area; the corruption, filled with dangerous flying enemies and gaping chasms; the jungle, with tall cliffs and swarming with powerful jungle bats and piranhas; and oceans at the end of the map. Directly below the surface there is the underground, accessible by finding a tunnel down,digging a shaft manually, eventually most players will be digging tunnels transverse from shaft to shaft. This under ground transitions from the near surface to the underground, and then a cavern layer. If the player keeps digging and following tunnels downward, they will reach the underworld, a hell-like area filled with the hardest enemies in the normal mode.
The player starts out with a minimum of equipment, an ax, a short sword, and a pick all in copper quality; however this is woefully inadequate for transversing the more difficult areas. In the first stage of the game the player tries to build a crafting base, then search for metal ore and items near the surface. Once the player has exhausted the starting area of goods, they will then plunge into the underground to find ore to forge weapons, tools and armor; to uncover useful items that cannot be forged but are none the less useful(such as the cloud in a bottle which allows the player to double jump); and very importantly the underground is where the player finds heart crystals that are broken to produce life crystals, which in turn when used increase the player's health by 20 points. The player will find rarer ores(from copper to silver to gold) more often and more frequent heart crystals the further down they go down into the underground, preparing the player for more difficult areas. After sufficient plundering of the underground the player is well equipped to go to the more difficult areas in the world; before this point the player would while technically able to progress through the more difficult areas(the desert is probably the most likly to be passable at the start) but would have been stymied by the strong enemies in combination with the difficult terrain; with the stronger armor and utility items the player is more able to survive and progress in the more difficult areas. Once the player has sufficient armor and health the first boss, Eye of Cthulhu, can spawn to fight them at night. Facing this boss is a sort of arbitrary signal that the player is ready to progress into the corruption and jungles, and should do so at this point.
This "second tier" after the ores of mundane metal is marked by the availability of new gear and ores to find. The player defeating the first and second major bosses causes demonite to be dropped in large amounts from the slain bosses; the second boss, the Eater of worlds, also drops shadow scales which is an ingredient for the Demonite armor, pickax and hammer. The process of summoning the Eater of Worlds to fight involves smashing shadow orbs buried at the bottom of the chasms in the Corruption which also creates the chance of summoning a meteor to strike. The meteor always hits a location on the surface away from the player, and transforms the area it hit into meteor ore, another second tier ore; to mine this ore the player need to have created a golden or better pickax. The Jungle is an alternative path that doesn't require a golden pickax, the player can use jungle spores from the underground jungle along with gems and stingers from dangerous hornets and vines from Maneater plant monsters to make jungle armor and equipment. Another possibility sans the pickax is that the player could go to the Dungeon, an area guarded by the third boss Skeletron who can kill instantly any player that enters the dungeon before defeating it at the entrance(yet he is still technically avoidable by a swift player with a sufficiently fast and agile character), find weapons in the chests that lay there and use the bones dropped from the skeletons as materials for Necro armor. However Skeletron is a strong boss, and this creates encouragement for the player to find other second tier gear first. The player can transverse all the way to the underworld and mine hellstone and obsidian to make hellstone armor and equipment; however this requires a nightmare pickax(this is the demonite pickax). After gathering sufficient gear that the player wants to progress, they then need to face the final boss of the normal mode. This boss, the Wall of flesh, is very difficult and requires either a large amount of preparation to defeat; the wall of flesh scrolls over to the player who must stay ahead of it or face certain death, and the Underworld consists of outcroppings on rock poking out of lakes of lava, so the player need a plan of travel to win. There are many solutions to this, drinking obsidian potions for lava invulnerability, somehow jumping from outcropping to outcropping usuing equipped mobility items, or the simpliest solution which is simply constructing a bridge long enough to keep the fight a straightforward running battle. The victory over the wall of flesh causes the world to enter Hard mode, with new equipment and ores avalible.
In Hard mode a third tier of ore can be mined, cobalt, mithril, and adamantium. The progression here is simplified from that of the second tier, and like that of the first tier, cobalt ore can be mined only by a Molten pickax(this is the Hellstone forged pick ax) or better; the mithril by a cobalt drill or better(for the third tier drills replace pickaxes), and adamantium by a mithril drill or better. Also in hardcore mode souls now appear as a drop from enemies, these are required to make the game's best equipment(for armor and weapons this is exalted equipment, and are forged also using cobalt, mithril, and adamantium versions of the resulting item to be forged). The souls required for most of this are dropped from three hard mode bosses, each a powered up form of a previous normal mode boss. The player only has to forge the summoning item for a boss and then use it in hard mode, and can fight the bosses as soon as they have the required materials to make the summoning items. The limiting factor here is that the bosses are immensely difficult, and forging third tier equipment is very useful and would a benefit to any player that takes the time to gather up the stronger armor. It's also likely that any player that is aiming to forge exalted equipment is already forging equipment from cobalt, mithril and adamantium.
It's worth noting here that there is very little "one way" game design decisions here, and what ones there are usually enforced by in game forces and not invisible programming walls. Entering the dungeon before defeating skeletron isn't impossible, but causes him to chase the player in an attempt to kill them. There's also the difficulty gates in the form of stronger enemies, as the player's stats outside of HP and MP is determined by equipped items that make players searching for greater stats search for better materials to make equipment. This is in turn limited by what tools the player has, starting with nightmare ore requiring a golden pickax; the player can sequence break but must do so with strong limitations on their equipment and items as a result.
This post written with the help of The Official Terraria Wiki to correct anything I may have misremembered