Evolving, like in the first Mystery Dungeon games, you are unable to evolve until you have reached a certain point in the game, in Darkness & Time's case, this is a requirement that needs to you beat most of the game. You have to follow the story beyond the "end" credits through to get the mission featuring Manaphy. After you have done that, unlock the Marine Resort Dungeon and you will be able to evolve at the Luminous Springs. These open up normally during the game and is found at the end of the dungeon listings
When you get there, you get a list of the Pokémon you have that are capable of evolving. The requirements are the same as in the main games so some require items in order to evolve. Select the Pokémon you wish to evolve and then it'll step into the springs. An animation will occur and your Pokémon will evolve. You can evolve multiple Pokémon at once so you don't have to go there hundreds of times
When you first get the evolution option open, you will get notifications that your starter and partner cannot evolve. This is true to an extent, however when you have completely finished the storyline of the game (even the bits after the credits) with the icons as seen below, and have recruited Palkia, you will be able to evolve your character and your partner, so do not fret if you cannot seem to evolve it

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