Let's see..
Find sand
Drop sand on slimes for gel
Make torches
Go underground, find bombs
Enter Corruption, smash 3 shadow orbs, fight Eater of Worlds on the surface
Damage EoW with the use of sand and the luck of fallen stars
No defeat necessary, just need enough for a nightmare pick, maybe armor for defense
Enter jungle, try to find Lihzahrd Temple
Mine any traps that may have spawned before the door. If no traps, use one of the exploits to get yourself through the door and get some traps
Find a hive and mine some hive blocks and scoop up a block of honey
Find an underground boulder trap and replace active/inactive stone blocks with spiky ball/spear traps
Summon Skeletron and lead him there, letting the traps finish him off
Enter dungeon, save mechanic
Mine dungeon bricks
Defeat goblins with traps, rescue goblin tinkerer
Reforge accessories and try to get good defense
Go to underworld to mine hellstone and obsidian for molten pickaxe
Make runway made of dungeon bricks
Fishing optional, try to get a lot of crates to open in hardmode
Set up wire stuff and use explosives to kill WoF. Might be tricky, but should be possible if you have enough money
Arrange lihzahrd traps to use against mechanical bosses. Need to wait for natural spawn
When warning message appears, box yourself up, place a hive in your box and mine it, repeat until a bee spawns. Bee should do constant 1 damage if you have enough defense. Place honey in your box, then place heart lantern and campfire nearby to heal back bee damage
Sit there as traps damage the mechs while they can't harm you as bee is constantly hurting you. Must have bee spawned before mechs appear
Craft drax/pickaxe axe at mythril/orichalcum anvil made from ores from crates or from mining up through the tiers, go mine chlorophyte
Set up traps and use bee exploit again to take down Plantera
Set up grinders for your desired endgame armor
Use a grinder for moon events and/or for Duke Fishron
Guess you're pretty much done then. Could replace bee exploit with slime statue exploit, but hives are usually easier to find. Not sure if you would count minions as weapons or not, but if allowed, use minions as soon as you can get them.