There are a few different ways to evolve your Pokemon. Each though depends on how they'd evolve in game. Here are the ways the can evovle and how it'd work here in Pokèmon, Go!

By leveling = This means at a certain level they can evolve. Such as Squirtle evolves into Wartortle at level 16. Meaning you simply gain them levels.

By Stones = This means they have to have a certain stone to evolve. Such as giving an Eevee a Thunder Stone will turn it into a Jolteon. So, you simply rp giving them the stone. Of course you must have one in your item bag to do this.

By Happiness = This can be slightly tougher. The Pokèmon's happiness must be high enough for it to evolve. Such as Togepi needing to be happy before it turns into Togetic. To do this you need to rp your trainer interacting with said Pokèmon and you can also bring it to the Pokèmon Pound and Resort Spa.

Night/Day time = This means that your pokemon needs to have a have happiness at either night time or day time to evolve. Such as Eevee can evolve into Umbereon at night time or it can evolve into an Espeon in the day time. You must simply rp or bring them into the Pokèmon Pound and Resort Spa at either day or night time hours.

By Trading = This has been changed in order to make this easier for you. In order to make Pokèmon that'd normally evolve by trading you simply need to rp them with your trainer and or bring them by the Pokèmon Pound and Resort Spa.