Yes, you need to level them up to evolve them. Once it hits a certain level you'll see a button you can press by it on your profile. This will evolve it.
Does this include Pokémon that normally require stones or trading to evolve? Do they need to reach a certain level or do you need to use an item (probably bought from the LV shop)?
My Growlithe is my highest leveled pokémon at level 10 and you'll probably heard more about its growth as I post.
I just got my buddy not too long ago, so my Alolashrew is only at level 4 xD Wait... nope, this post leveled it up to 5.
My highest would be my level 5 Growlithe. Posting to level them up was definitely a good idea. I think this is the most enthusiastic I've been for posting outside of Postfest. I can't wait til I can evolve my 'mons!
My Alolan Sandshrew is at level 15 right now. I feel so happy knowing that they're going to be evolving soon!