Your Pokemon are only damaged when you take part in Gym fights at an enemy team's gym, Or doing training at a friendly Gym.
Your Pokemon will faint if you lose them when fighting an enemy gym (requiring you to revive them).
You Pokemon will only go down to 1 health if you do training with them (so they can just be healed)
Pokemon do not heal naturally over time , which is a shame, but they also only take damage when you do a Gym battle. But below are the ways i have found to heal them.
There are 3 ways to heal your Pokemon:
There are 2 ways i've found to revive your Pokemon
You get these potions as rewards from pokestops. Its random chance whether you get pokeballs, potions or a mix of both.
The items you get from a pokestop are based on your level, so you will only start getting potions once you get above level 5.
You also get some when you level up at certain levels, but that takes time obviously.