Rot Boxes are the only capture tool in the game. Higher tier boxes improve catch rate and raise the level cap, so upgrading in order is the most coin-efficient way to expand your party.
In the Grass Zone, start with the Rot Box at 31 coins (handles up to Level 9). At Level 3, buy the Silver Box for 197 coins (extends to Level 15). At Level 6, upgrade to the Gold Box for 401 coins (handles up to Level 22), which carries you through most of the Grass Zone.
When you transition to the Ice Zone, the box meta resets. The Snowbox (681 coins) is the entry point. The Snowman Box bridges mid-game, the Miner Box (1,680 coins) is the steepest single upgrade, and the Frozen Box (2,250 coins or 140 Robux) is the practical end-game option for late Ice Zone encounters.
Both the Grass Zone and Ice Zone have their own Rot Centers at the main hub. Walk in, talk to the NPC, and the box list will show every option available at your current Player Level.