Evolve on Pokémon Mystery Dungeon Blue Rescue Team

Here's a nice and fairly brief overview on how to evolve in Pokémon Mystery Dungeon Blue Rescue Team.

Steps

Start as a basic Pokémon. As you progress through the game a Lombre in Pokémon square will tell you you can not evolve. Sure enough you will find this out once you reach level 15/16 depending on which Pokémon you are.
To level up you must first beat the boss of the game when you defeat it, you must go back to Pokémon square and up to Whiscash pond. Where you could once stand and talk to Whiscash is now a massive hole in the ground, this hole leads you to an underground cavern called the luminous cave. In here, you will be able to evolve your Pokémon.

Go to the path leading to the friend areas. Once there you may be able to find out whether or not your Pokémon will be able to evolve. To do so you must open up the status window of the Pokémon selected it will tell you if it can or can not be evolved.

Some Pokémon require items to evolve such as Pikachu needs a Thunderstone and Eevee needs a stone. Some Pokémon may also need the link cable which you can find randomly in dungeons. Others may need increased stats plus ribbons, the next section tells you all the items/stats you will need.
You may only take in one Pokémon to luminous cave so the max numbers of party Pokémon may only be one. You learn how to do this as the epilogue starts and you're about to do the first epilogue mission.