The short tl;dr walkthrough

Chop 150-250 wood to build a rectangle about 1/3rd to 1/2 the width of your screen. Craft a workbench, doors, a table, and a chair. Kill slimes for gel to make torches. Use wood to craft wooden armor, a wooden broadsword, and a wooden bow. Use stone blocks and wood to craft arrows for your bow.

Explore left or right to find natural caverns and openings. Explore them to find pots and chests for materials. Look for Life Crystals, mine them with your pickaxe, and use the resulting hearts to increase your HP.

As you increase your HP, mine ore to create bars, which are used to craft equipment and higher-level pickaxes, axes, and hammers.

Periodically dig a vertical tunnel straight downwards, called a Hellevator.

Get 10+ defense and 200+ HP to fight the Eye of Cthulhu, your first boss.

Start building homes for NPCs in their preferred biomes and have them move in to build up your NPC (non-player character — anyone the game controls, not you) towns.

(Optional, highly recommended) If a Goblin Army event occurs, complete it. A Goblin Tinkerer NPC will spawn underground. Find and rescue him by talking to him; he offers useful items and the Reforge option.

Go into your evil biome and break three Shadow Orbs or Crimson Hearts to spawn your second boss: Brain of Cthulhu for Crimson worlds, or Eater of Worlds for Corruption worlds.

(Optional) After defeating your evil biome boss, meteorites may spawn. Mine them to make meteorite bars, which are used to craft meteorite armor and the Space Gun, which is quite powerful.

Explore the surface until you find the Dungeon. Talk to the Old Man NPC to spawn and kill Skeletron.

Continue exploring underground until you have 400 health.

(Optional, recommended) Explore the now accessible Dungeon. Kill enemies and break pots to find Golden Keys to open Golden Chests. Try to find a Shadow Key for the Underworld.

(Optional) Explore the Jungle biome until you find a Hive mini-biome. Build a small arena and break the larvae jar to spawn the Queen Bee boss.

(Optional, highly recommended) To prepare for Hardmode, build five-tile wide tunnels around your Jungle (and optionally your forest biome) to prevent the Corruption/Crimson and Hallow biomes from spreading and ruining them. Place wood on the sides to prevent biome spread.

(Optional, highly recommended) Get the Witch Doctor NPC and make a Jungle home for him.

Go to the Underworld via your Hellevator. Build long stretches of wooden platforms to create a lengthy runway. Use your Shadow Key (if obtained) to open chests and find the Hellwing Bow. Find and kill a Voodoo Demon to get the Guide Voodoo Doll. Drop it in lava to spawn and kill the Wall of Flesh boss.

(Optional, highly recommended) Go to your Witch Doctor NPC and buy Leaf Wings for 1 platinum. Wings are very important for hardmode bosses.

If you did not get Leaf Wings, kill Wyverns in the Sky layer for 20 Souls of Flight and kill Harpies for a Giant Harpy Feather. Craft Harpy Wings.

With your Pwnhammer, go into your evil biome and break three altars to spawn three tiers of new ores. Break a total of 9 altars for an easier time finding veins.

Mine the first tier of ore and make a pickaxe. Mine the second tier of ore, make a pickaxe and an anvil. Mine the third tier of ore, make a pickaxe and a forge. You can now craft all Hardmode items.

Get the Destroyer summoning item or wait for it to spawn naturally. Kill it.

Get the Skeletron Prime summoning item or wait for it to spawn naturally. Kill it.

Get The Twins summoning item or wait for them to spawn naturally. Kill them.

Go to the Underground Jungle and look for little pink bulbs. Build a large vertical arena around one. Mine it to spawn Plantera.

Find the Lizard Temple in the Underground Jungle. Use the Temple Key to enter, reach the end, and spawn and kill the Golem boss.

(Optional) Go to the Cavern layer and find an Underground Mushroom biome. Use your Bugnet to capture a Truffle Worm, or build a Mushroom Biome in the Cavern layer for them to spawn. Fish with it in the Ocean biome to spawn Duke Fishron. He can drop nice equipment but is very difficult.

Go to the Dungeon and attack and kill the Lunatic Cultists. Pick up the Ancient Manipulator it drops to craft celestial equipment using materials dropped from the Celestial Pillars.

Four Celestial Pillars will spawn. Kill 100 enemies (150 on expert) for each pillar to deplete their shields and kill them. Collect the fragments.

After all four Celestial Pillars are destroyed, a one-minute timer activates, and the Moon Lord, the final boss, spawns. Use Celestial Pillar fragments to craft the Moon Lord boss summon item.