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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.