I have always wanted to answer this question but I find myself not following the 'Ideal' progression of the game by skipping some parts - especially pre hardmode. It is only by now I have finally played enough to have enough experience to write a complete answer I can be satisfied with. This will be a very long answer but I believe it will also serve as a decent walkthrough for new players(are there still any?) and also a checklist for existing players to find new things to do in the game.

Since this is a very long answer I'll not link every item/monster in game to the Wiki since I'll be effectively linking to every page of the Wiki. Instead, I'll simply quote them like this `Gel` for names that can be searched in the Wiki for more information.

 1. Cut `Wood` from nearby trees, collect `Gel` from `Slimes`. Build `Workbench` to craft basic items including Wooden equipments(Armour, Sword and Bow). Combine `Wood` and `Gel` to make `Torches`. *Each type of `Wood` depending on its source crafts into a different set of armour and weapons that provides different stats* 
 2. Dig nearby `Stone` to make `Furnace`(need `Torches`) and Arrows(need `Wood`).
 3. Build basic housing(No opening, Walled and have Chair, Table and Lighting) *Stone can be made into `Grey Bricks` at the furnace to prevent the house from being corrupted or hallowed*
 4. Explore sidewards within safe biomes(Forest and Snow), break all pots and open `Chests` to get basic essentials for exploration(Potions, Ropes etc.)
 5. Explore surface caves for more basic loots. Dig out any ores found to make better Weapons/Armours. If not enough ores are found to make equips for next phase, dig slightly deeper to find natural Caverns for more ores.
 6. Build arena in anticipation for `Eye of Cthulhu` - Spawns after first player have 200 or more health.
 7. Explore surface/Caves of tougher biomes(Dessert, Jungle, Crimson/Corruption) for biome specific loots. *Dessert's most important loots are inside Pyramids that are sometimes not visible from the surface, Jungle have the most `Crystal Heart` which increases max health also can obtain `Ivy Whip` for exploration and Jungle Armour for mages, Crimson and Corruption have `Crimson Heart` and `Shadow Orb` which 2 can be safely broken for loots before the third summons the respective biome boss*
 8. If `Gravitation Potion` is obtained use it to find Sky Islands, otherwise leave this for later or build a Skybridge(a map spanning horizontal platform in high altitude).
 9. Fight `Brain of Cthulhu`/`Eater of Worlds` and Skeletron.
 10. Explore Dungeon and Jungle Underground for loots.
 11. Build Hellevator. *a straight tunnel from surface that ends right before hell with a pool of water 2 tiles deep*
 12. Fight `Wall of Flesh`. *Build a horizontal bridge across hell to make fight easier* - **Entering Hard Mode**.
 13. Break Altars with `Pwnhammer` obtained from `Wall of Flesh`. *Better hammers obtained in Hard Mode can also break the Altars*
 14. Explore Ceiling of Hell and deep underground for Hard Mode ores. *Also available closer to the surface but usually more deeper underground*
 15. Explore Hallow and Crimson/Corruption especially underground for loots. *Souls of Light and Night will only drop underground, you will know you are deep enough when different monsters in that biome spawns also you will see `Crystal Shards` if you are in Hallow*
 16. Hunt `Wyverns` in high altitude for `Soul of Flight` for wings.
 17. Hunt in Blizzard and Sandstorm for rare monsters with unique loots.
 18. Fight Mechanical Bosses.
 19. Explore Underground Jungle for `Chlorophyte Ore` and `Life Fruit`
 20. Fight `Plantera`.
 21. Explore Dungeon for loots *parts with different wall will spawn different monsters and each type of wall have its unique loots*
 22. Find and explore Jungle Temple in the Underground Jungle. *The traps are potentially more deadly than the monsters*
 23. Fight `Golem`
 24. Kill cultists at Dungeon entrance to Spawn `Lunar Cultist`, kill him(honestly I am not sure if it's a 'him'). *Everything from here on is the 'End Game' phase*
 25. Fight the 4 Lunar Pillars
 26. Fight Moon Lord *spawns automatically after last pillar is destroyed but with a delay so enough time to teleport home and heal/buff before the fight starts, alternatively can be spawned from `Celestial Sigil` crafted from the Shards obtained from the pillars*

These are events/bosses that are not within the main progression and can be skipped/done later.

**Pre Hard Mode**

 1. King Slime - Spawned from `Slime Rain` event which occurs randomly after more than 150 slimes are killed and a player have more than 140 hp plus 8 def.
 2. Queen Bee - Spawned by destroying Larva found within bee hives that naturally appears in Underground Jungle
 3. Goblin Army - Randomly occurs after a player have more than 200 hp and the world have at least one `Crimson Heart`/`Shadow Orb` destroyed
 4. Old One's Army - Requires a player to place a `Eternia Crystal` on the `Eternia Crytal Stand` in a valid location. Both are purchasable from the `Tavernkeep` npc

**Hard Mode *Incomplete***

 1. Duke Fishron - Spawned by fishing in the Ocean with a `Truffle Worm` caught in Mushroom biome.
 2. Pirate Invasion - Started by using `Pirate Map` obtained from kills near/in the Ocean Biome
 3. Frost Moon - Started by using `Naughty Present` at night
 4. Pumpkin Moon - Started by using `Pumpkin Moon Medallion` at night
 5. Solar Eclipse - Chance of naturally occurring each day after Mechnical Bosses are defeated also start by using `Solar Tablet` in the day
 6. Martian Madness - Started by allowing a `Martian Probe` to spot player and flee.
 7. Old One's Army - Changes once after Mechnical Bosses are defeated and once more after Golem is defeat to become a harder event with better loots.

*Apart from Solar Eclipse all other bosses/events can be avoided by reseting the world(host leaving and rejoining or server reset) Solar Eclipse can be avoided by going deep underground*

**Things that can be done to make game easier/more interesting**

 1. Dig a horizontal tunnel right under the surface and lay it with `Asphalt Blocks` for traveling quickly across the map
 2. Build a mine cart track that reaches into each biome
 3. Build Teleporters that allows instant access to each biome *The first three points are basically phases of fast traveling in the game*
 4. Create Mixed Biomes by placing large amount of different biomes tiles within the same area. *Useful for Angler quests and hunting monster drops*
 5. Purify the world! *I'd respect anyone who can do it with `Purification Powder` but I'd recommend doing it with a `Clentaminator` that the steam punk sells*

I have made my answer a community wiki so please feel free to expand on each point or even add in anything I missed.