After You is a move that's specifically for double battles. It basically forces the target to move immediately after the user, so if you use a very fast pokémon with After You, and target your slow teammate, you can basically make that teammate go much earlier in the turn than normally. Lilligant is a pretty popular user of After You in Sun teams as it's really fast with the Chlorophyll boost.
Me First is a move that only works if you move before the target, and only if the target uses a damaging move that isn't a Z-Move. It uses the move that the target intends to use and boosts the power by 50%. It's a situational move that can catch opponents off-guard, for instance by hitting an Alolan Marowak intending to go for Shadow Bone, or a Latios going for Draco Meteor, for super effective damage, but it requires a lot of good prediction to pull off, as you simply waste a turn if your opponent is faster, goes for setup, or uses a Z-Move. Some fast users of it are Mega Absol and Hawlucha.