Munchlax and Happiny — Route 1
Munchlax, baby form of the formidable Snorlax, can be caught extremely early in the game: in the patches of grass directly to the right of your house on Melemele Island. It’s an exceptionally rare encounter, but it’s in there.
Happiny, unevolved form of Chansey, can also be caught on Route 1, but only during SOS battles (battles where the Pokémon has called for help). Rumor has it wild Bonsly are most likely to summon a Happiny to help.
Bagon — Route 3
The dragon Pokémon that eventually evolves into the fearsome Salamance can be found more easily later in the game, but you can run into it very early on Melemele Island (the first island). There’s a single patch of grass on Route 3, to the right of the bridge that leads back to Route 1. Its encounter rate is insanely low, but if you’re determined to have a Salamance on your team right at the start, it is possible.
Passimian/Oranguru — Lush Jungle
The twin (fraternal, not identical) new monkey Pokémon can be found early on, on the second island, in Lush Jungle. Passimian only appears in Sun, and Oranguru in Moon, and they’re both extremely rare encounters in their respective games. All you can do is keep hunting.
Feebas — Brooklet Hill
When it was introduced in generation 3 (Pokémon Ruby and Sapphire), Feebas was insanely hard to catch. Now it’s just a matter of beating the odds while fishing at Brooklet Hill. Look for fishing spots with bubbles coming out of them to improve your odds of finding the weird fish that evolves into the beautiful Milotic.
Vulpix/Sandshrew — Tapu Village
Vulpix (in Pokémon Sun) and Sandshrew (in Pokémon Moon) have been around in Pokémon forever, but in Alola they get new typing and they look awesome. They’re common on the path approaching the Elite Four, but you can actually get them much earlier than that: in the grass in Tapu Village on the third island, with a little patience.