Unlike previous games, evolution in Mystery Dungeon seems nothing more than to get certain Pokémon, in other words, to fill your Friend Areas with every single one of the 386 species. The interesting thing is all you get from evolution in this game is an altered appearance, and for some an additional type or ability to learn different attacks, but evolution does not raise your stats like it drastically does in previous games.
To the Pokémon inhabitants in Mystery Dungeon, evolution seems to be a new concept to them as well, as it is unheard of by the characters until Rayquaza has stopped the meteor, so evolution is in other words impossible before you have beaten the first half of the game and seen the credits roll.
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 evolutions.

Whenever you enter the Light Cave, the glowing stone will ask if you want to evolve, click yes, and it will ask if you want to evolve without items, or with items. Depending how your specie evolves, select accordingly. Note that you must leave the cave and come back again if you want to further evolve, provided that you are at the required level.
You can evolve your other Pokémon too by the 'Leader Switch' option. Go to that particular Pokémon's Friend Area, talk to it, first select the first option to make it join your team. After you do so, talk to it again and pick the second option which is the leader option, this makes that particular Pokémon your leading one.
Remember to send back the rest of your team to their Friend Areas (either go to their Friend Area to do that or by your menu, check the Main Menu section), and you may access the Light Cave for evolution.
The thing about evolution in Mystery Dungeon is that moves are only learnt when you gain a level, not the instant a Pokémon evolves. Take Charizard as an example. Normally it learns Wing Attack at level 36, but you can't get a Charmeleon to evolve until it is at level 36, so when you evolve it, Charizard is already at level 36 and evolution alone will not make you learn the move at that level.
There are two ways to make your evolved Charizard relearn Wing Attack in this case. One, there is an item called the Unfortunate Seed, eat it and your level will decrease by one. Say for Charizard again, you return to level 35 after eating one Unfortunate Seed, so when you level up to 36, you will get to learn Wing Attack.

The easier way will be to go to Gulpin's Connect Shop at the Pokémon Square, where move relearning to possible. Select the Pokémon of your choice, then the last option. You will see a list of its old move to learn from. Like when you choose to connect your moves, Gulpin charges for a fixed fee for move relearning.
Some Pokémon must be evolved with special items like in the previous games. In the Light Cave, when it asks you whether you want to use an item, click yes (second option) and you can select the item from your item bag.
After you do so, you will be asked again if you want to proceed with the evolution, or select another item. For Pokémon that evolve with one item, select proceed (first option).
Otherwise, attach a second item (second option) and it should automatically proceed to evolution.

The following special items or means are needed for certain evolutions:

Fire Stone - Fire Great Land

Base No. Pic Base Name Final No. Pic Final Name
#037 Vulpix #038 Ninetales
#058 Growlithe #058 Arcanine
#133 Eevee #136 Flareon

Leaf Stone - Wish Cave

Base No. Pic Base Name Final No. Pic Final Name
#044 Gloom #045 Vileplume
#070 Weepinbell #071 Victreebel
#102 Exeggcute #103 Exeggutor
#274 Nuzleaf #275 Shiftry

Moon Stone - Sun Cave

Base No. Pic Base Name Final No. Pic Final Name
#030 Nidorina #031 Nidoqueen
#033 Nidorino #034 Nidoking
#035 Clefairy #036 Clefable
#039 Jigglypuff #040 Wigglytuff
#300 Skitty #30 Delcatty

Sun Stone - Sun Cave

Base No. Pic Base Name Final No. Pic Final Name
#044 Gloom #182 Bellossom
#191 Sunkern #192 Sunflora

Thunder Stone - Lightning Great Land

Base No. Pic Base Name Final No. Pic Final Name
#025 Pikachu #026 Raichu
#133 Eevee #135 Jolteon

Water Stone - Northwind Great Land

Base No. Pic Base Name Final No. Pic Final Name
#061 Poliwhirl #062 Poliwrath
#090 Shellder #091 Cloyster
#120 Staryu #121 Starmie
#133 Eevee #134 Vaporeon
#271 Lombre #272 Ludicolo

Link Cable - in place of HMs or other items when you already have the item (Sun Cave, Flying Dragon Hole)

Base No. Pic Base Name Final No. Pic Final Name
#064 Kadabra #065 Alakazam
#067 Machoke #068 Machamp
#075 Graveler #076 Golem
#093 Haunter #094 Gengar

Link Cable + Deep Sea Scale - Big Sea

Base No. Pic Base Name Final No. Pic Final Name
#366 Clamperl #368 Gorebyss

Link Cable + Deep Sea Tooth - Big Sea

Base No. Pic Base Name Final No. Pic Final Name
#366 Clamperl #368 Huntail

Link Cable + King's Rock - Wish Cave

Base No. Pic Base Name Final No. Pic Final Name
#061 Poliwhirl #186 Politoed
#079 Slowpoke #199 Slowking

Link Cable + Dragon Scale - Flying Dragon Hole

Base No. Pic Base Name Final No. Pic Final Name
#117 Seadra #230 Kingdra

Link Cable + Metal Coat - Southern Hole

Base No. Pic Base Name Final No. Pic Final Name
#095 Onix #208 Steelix
#123 Scyther #212 Scizor

Link Cable + Upgrade - Wish Cave

Base No. Pic Base Name Final No. Pic Final Name
#137 Porygon #233 Porygon2

Smartness (four stars or above)

Base No. Pic Base Name Final No. Pic Final Name
#042 Golbat #169 Crobat
#113 Chansey #242 Blissey
#172 Pichu #025 Pikachu
#173 Cleffa #035 Clefairy
#174 Igglybuff #039 Jigglypuff
#175 Togepi #176 Togetic
#298 Azurill #183 Marill

Smartness (four stars or above) + Sun Ribbon - Flying Dragon Hole

Base No. Pic Base Name Final No. Pic Final Name
#133 Eevee #196 Espeon

Smartness (four stars or above) + Moon Ribbon - Northwind Great Land

Base No. Pic Base Name Final No. Pic Final Name
#133 Eevee #197 Umbreon

Beautiful Scarf - Western Cave

Base No. Pic Base Name Final No. Pic Final Name
#349 Feebas #350 Milotic

 
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