How to get the starters
WARNING: YOU NEED TO BE RANK X FIRST!

Chespin: Quest at the school in front of the taxi. Defeat thrice in a row for it to join your team.

Froakie: Quest at the scaffolding on the boulevard at Jaune Sector 8. Beat the obstacle course in 30 seconds OR LESS. Afterwards, you need to do another obstacle course, now with a time limit of 40 seconds. Then, the last obstacle course is all the way up at Rouge Pokémon Center. This one has to be beaten in 50 seconds. After this one, Froakie joins your party.

Fennekin: Quest at Magenta Sector 2. It is sick. Find a old herbalist at Cafe Classe(Cafe across Vernal Pokémon Center). After talking to him(he's marked as a Old Man), go to Wild Zone 5. Collect a Revitalizing Twig from the area you can find mostly just Venipede in, after following the water. The tree with the twig is on the elevated area you need to reach with the ladder. I recommend you use the quest guidance, since that makes this part of the quest VASTLY easier. After getting this twig, go back to the lady at the sick Fennekin. You will give the twig to the Fennekin and it will join your party.