Play through the story until you reach the point where a time portal in the sewer opens, allowing you to freely travel between the past and future.
Jump through to the past and talk to the NPC next to the portal. Equip the Jefferson costume and use its power to persuade him into giving you the mystery candy. Plant the mystery candy in the pile of dirt just above him.
Return to the future and find the gardening NPC named Dirtonian in the Dental Compound area. Use the Jefferson costume power on him as well. He wants you to help him remember a movie he saw once. You need to equip costumes related to the hints he gives you.
Equip the 3 following costumes:
Upgrading these three costumes is also required. Upgrades can be purchased from Shady's secret shop. Once you have them all, talk to Dirtonian until he gives you the fertilizer.
Next, return to the past and use the fertilizer on the planted seed. Go back to the future again and you'll see it's turned into a massive tree. Go through the research lab stairs and find the entrance into the tree. After ascending to the very top, you'll find the complete Solar System costume in a chest.