Rescue Teams Blue/Red:
After the credits have rolled, a part of the shattered meteor has
fallen into the Whishcash Pond north of the Pokémon Square and created
the Light Cave. It is where all your evolutions take place. However,
you can only enter the Light Cave if you are on your own, which means
your partner has to go, but when? After you have tried in vain to
visit the Light Cave once, and perhaps obtained the Dive HM from
Whiscash, when you return to your headquarters, your partner will tell
you that it may not be helpful sticking around all the time since you
will need the help of other companions, and it will return to its own
Friend Area. Now, you may enter the Light Cave and start your
Your Pokemon will evolve after the game if you go into Light Cave.
Time/Darkness/Sky: You evolve Pokemon in Luminous Springs, which is unlocked when you defeat the Guild members in Mystifying Forest. When you graduate, regular Pokemon can evolve there, but not the two main characters. You can start to evolve the two main characters only once Manaphy has told you about Marine Resort.
Luminous Spring, Bulbapedia
Gates to Infinity:
The player starts out as their partner Pokémon for several days until they complete the Worldcore and make a wish for their playable Pokémon to return. The player decides whether to return or not and evolution may not happen until they return to Pokémon Paradise. After the player returns to Pokémon Paradise and completes one dungeon, both the player and partner Pokémon may now evolve if the player chooses to evolve them inside of dungeons.
Your Pokemon will evolve inside of dungeons at a certain level.
After Main Story Line, Bulbapedia