Any Pokemon you want healed, you leave at a Pokecentre, and the healing won't be instant. If it's only HP then generally it won't take very long. But other things such as injuries/illness can take a while.
It may seem like a put-off, having to wait, but when you consider the fact that Trainer Battles are refuseable (and thus you don't have to go wading through a small army of them on route to the next town) then its not such a nuisance.
The previous two screenshots have been showing the Live Update part of the Laptop. This allows you to check on the pokemon you've left in Dojos and Pokecentres, and see a rough idea of how long it'll take until they've finished their training/are fully healed.
That's what the green bars represent in the last screenshot. It's going to take quite a while for Pidgey's wing to heal.
It also tells you in which Pokecentre/Dojo you've left the Pokemon. Again, this may seem like a nuisance if you left your Pikachu all the way back in the first Pokecentre you came across, but you also have to take into account that you can Fly from the beginning of the game - well as soon as you catch a Pokemon capable of carrying you anyways.