Walkthrough

1. The first task in the Initiation Challenge requires you to gather 5 Leppa Berries. These are abundant almost everywhere, especially in Withered Wasteland.
2. Simply find a tree, water it if it’s dry, and you’ll be able to get Leppa Berry trees. Headbutt them to collect the berries.
3. Completing this rewards you with the Bouldery Badge.
4. Withered Wasteland Completion Guide
5. Next, plant the following crops and gather 10 of each. You can get these locally in the areas mentioned, but to replant them, you'll need a Rototiller to plant them and the Grow specialty to speed up their growth. Completing this rewards you with the Cascade-like Badge.
6. How to Plant Seeds to Grow Crops
7. The next part requires material gathering and the use of Pokemon specialties. For the Lumber, cut down trees to get small logs and give them to Pokemon with the Chop ability to get Lumber.
8. For the Paper, give Wastepaper to Pokemon with the Recycle ability to get it.
9. Lastly, Fluff can be dropped by Pokemon with Litter. Specifically, you can befriend Mareep or Jumpluff, and they'll drop these materials around their habitat or house if you rehomed them. Completing this rewards you with the Thunderish Badge.
10. For the next step, gather Squishy Clay and ask Torchic to burn it into Bricks. Squishy Clay can be obtained by using Rock Smash on Clay Blocks. It may also be dropped by Pokemon like Paldean Wooper and Clodsire.
11. Gold Ingots can be obtained by converting Gold Ores in the Smelting Furnace. You can find Gold Ores in Rocky Ridges, or use the Arcanine Doll and visit the Dream Island if you're short on it.
12. Concrete, on the other hand, can only be obtained after unlocking Sparkling Skyland. You'll need a Concrete Mixer, Limestones, and Pokemon with Crush, like Onix. Gathering all these rewards you with the Rainbowish Badge.
13. The next step requires more work compared to the previous ones. You'll need to build electric sources, use the Furnaces again, and gather several materials to progress.
14. The easiest step is perhaps gathering the Crystal Fragments, which are easily available in the Rocky Ridges Mines. Completing this rewards you with the Soul-like Badge.
15. To generate 50 Electricity, simply make two electric sources. The easiest ones you can build are either a Furnace or a Windmill, so build at least two. One source generates 30, so to reach 50, you'll need another one.
16. Tinkagears can be obtained by asking Tinkmaster to convert Iron Ingots into Tinkagears. Tinkmaster is only accessible once you reach Sparkling Skylands.
17. Iron Ingots can be obtained the same way as Gold Ingots, except you need Iron Ore. You can get Iron Ore by asking a Pokemon with Recycle to convert nonburnable garbage into it, or by breaking any Rusty Blocks you find.
18. How to Get Tinkagears
19. The next step focuses on crafting several pieces of furniture. You only need one of each, and the recipes and materials for these are easily obtainable as you progress through the game. You can get the Marshy Badge from this.
20. Crafting Guide: How to Craft Items
21. The next step may also involve crafting, but if you already have these items, feel free to submit them since you get all of these for free. By raising the environment level to 5 in Bleak Beach, Rocky Ridges, and Sparkling Skylands, you get these items as rewards. Completing this step rewards you with the Volcanoey Badge.
22. How to Raise Environment Level
23. Perhaps the easiest step of this request is to submit a photo of your choice. This can be any photo you took with the camera and saved in the album!
24. Completing this step gets you the Earth Badge.
25. Finally, craft two Party Poppers to complete the quest. These can be obtained by raising the environment level to 5 in Withered Wasteland, but the recipe can also be bought in the PC.
26. You'll get the last reward of the line, the Memories Photo Frame, for completing this.
27. Gather 5 Leppa Berries Reward: Bouldery Badge
28. Gather 10 Beans, 10 Tomatoes, and 10 Wheat Reward: Cascade-like Badge
29. Gather 20 Lumber, 5 Fluff, and 10 Paper Reward: Thunderish Badge
30. Make 40 Bricks, 20 Gold Ingots, and 50 Concrete Reward: Rainbowish Badge
31. Generate 50 Electricity, Gather 10 Crystal Fragments and 5 Tinkagears Reward: Soul-like Badge
32. Craft 4 Industrial Beds, 4 Resort Lights, and 4 Office Desks Reward: Marshy Badge
33. Obtain a Refrigerator, Game Boy System, and a Washing Machine Reward: Volcanoey Badge
34. Submit your Favorite Photo Reward: Earth Badge
35. Get Party Poppers Reward: Memories Photo Frame

Walkthrough source: Game8.