I'd probably not change too much, just some of Ash's lineups. For example:
Ash would get an Eevee in Celadon City, which would later evolve into an Espeon. Ash would also keep Haunter, and since Pokemon don't necessarily have to be traded in the anime to evolve (Poliwhirl-Politoed), it would evolve normally.
All would stay the same
Cyndaquil would evolve just before the Silver Conference. Totodile would evolve during the Whirl Cup, and Croconaw would evolve during One Trick Phoney. Eevee would evolve a few episodes after meeting Gary's Umbreon, ultimately losing to it in the League.
Phanpy would sneak onto the ferry for Hoenn, appearing as a target for Pikachu to release its excess electricity due to its immunity to Electric attacks. It would evolve into Donphan during... eh, let's say Vanity Affair against Drake's Shelgon and be released in Date Expectations.
Also, during his trip to Hoenn, Brock would catch a Vulpix of his own (since he never considered Suzy's Vulpix to really belong to him) that would become his Pikachu for Hoenn and beyond. Other than that, everything's the same.
Rotating sixth member
Corphish would evolve through training after the first Anabel battle. With its new Dark-type, it would star in the battle against Anabel.
The gang would've met Bertha earlier than the Draco Meteor episode. She would have a gible that takes a kind of liking to Ash. After Gliscor loses to her Hippowdon, she would propose a trade - her Gible for Ash's Gliscor - for the sake of training Gliscor and the happiness of Gible. From there, Gible would struggle with Draco Meteor like normal.
Also, Jessie would catch an Eeeve and evolve it into a Jolteon, which she would tout as Team Rocket's ultimate defense against Pikachu (at least until Ash gets wise to her plans).
Pokemon Y Friend Code: 3282-3535-2249
Friend Safari: Pansage, Petilil, Gogoat
As of now, my focus is on Safaris with starters (barring Wartortle and Quilladin), but give me your Safari anyway. Chances are I'll find something interesting.