Veterinary is a wonderful skill. Standing next to your pet and slap bandages on costs next to nothing, and heals more than magery. Using veterinary can make hunting stronger monsters possible with significantly weaker pets.
Veterinary is a lot like healing. It requires animal lore, much like healing requires anatomy. Animal lore will raise passively while using veterinary, just like anatomy raises passively while using healing. Applying a bandage will take approximately 3-5 seconds, depending on your dexterity (5 seconds at 39 or lower dex, 4 seconds at 40-99 dex, 3 seconds at 100 dex). You may not think it just by looking at it, but one second makes a huge difference in how fast you can heal. You must be within 1 tile of the pet while applying the bandage and when the bandage completes the application process. If the pet walks away after applying the bandage, it will not disturb the healing process, as long as you return to the pet's side before the application finishes.
To cure a pet with bandages, you will need to have a minimum of 60.1 veterinary and animal lore. Curing stronger poison is more difficult than weaker poisons, and will require higher skill to do so. To resurrect a bonded pet, you will need 80.0 veterinary and animal lore. Only bonded pets can be resurrected. Resurrecting a pet will result in a .1 penalty to all skills.
Veterinary raises in the same way that healing does. While attempting to heal 75% of a pet's health is more difficult than healing 50% or 25% of the same pet's health, all 3 levels of health have the same chance of yielding a skill gain. Curing and resurrecting are completely difficulty based when it comes to skill gain. If your skill is high enough, curing a low level poison will not be challenging enough to yield a skill gain, but a higher level of poison might. For resurrecting and curing (depending on the poison level), there will come a point in time where you will have 100% chance of success, which means that you will no longer be able to gain skill from those actions.