How to Get Rocket Boots in Terraria

Defeat a Goblin Army. To do so, have at least one player with 200+ health and have smashed at least one Crimson Heart or Corruption Orb. Once both requirements have been met, there is a small chance of a goblin battle arriving every morning at dawn (1 in 30 pre-hardmode, 1 in 60 hardmode)

After defeating the Goblin Army, find a cave and start going down it.

Keep on going down, until an NPC bound in ropes appears. Untie him, (by talking to him) and talk to him.

Select the "Rocket Boots" and buy them. (for 5 gold coins)

If you have made a house that is habitable by an NPC, he will spawn in, and live there at your convenience.

Don't worry about any bosses spawning, except the Eye of the Cthulhu. It may spawn if there are at least 3 NPCs in the world, including the Old Man, and if the Eye has not been yet defeated in that world. (This cannot happen during the day).

To upgrade your rocket boots even further, you may use a Tinkerer's Workshop (Also bought by the Goblin Tinkerer), to combine the Rocket Boots with either Hermes Boots, or Flurry Boots to make the Spectre Boots. Even further, you can make Lightning Boots, and even Frostspark Boots!