Follow the stops in order, island by island through the trials and grand trials to the post-game. Each Pokémon is listed at the first point you can reach a place where it appears. Surfing and fishing spots count from Lana's trial, when you get the Lapras ride and a fishing rod. Tap a row to check it off. Evolves into shows later stages and how they evolve.
Shiny Charm: catch every Pokémon in the Alola Pokédex (Magearna, Marshadow and Zeraora don't count), then talk to the game director at the Heahea City hotel. You'll need trades for the other version's exclusives.
Totem Stickers
The trial captains have hidden 100 Totem Stickers around Alola. They glow, and some are tucked behind objects or Pokémon. After your first 20, Samson Oak calls you, and from then on he gives you a Totem-size Pokémon at each milestone below.