First off, great question.
Method 0: Most people will say: "skill ball it up to 120", but if that's not an option--or your style-- and you actually like play Ultima Online, I suggest Method 1 or 2.
Method 1: set a macro in the elemental training room, to use the Veterinary skill on your pet, while it is attacking the training elemental. Because it is legal to macro skills-- on the EVO shard-- your Veterinary skill will raise over time. Note: just because you can leave this afk unattended, doesn't mean it is a good idea for long periods of time. Pets can go wild, and kill new players accidently, or you may reach a threshold where Staff will consider your pet a problem/idle traffic and delete it from the server. I suggest checking in on your pet every hour at the minimum, and leashing your pet before you go to sleep or work for the day to be safe.
Method 2: tame a random pet and bond it to your character. The pets level doesn't matter, since the pet will be a throw away, after your training is over. Once the pet is bonded, slay the pet-- I know 
-- and attempt to resurrect it with a bandaid. For best results set a up a macro to continuously attempt to resurrect the bonded dead pet, but make sure to skip the part where you actually resurrect it. This is the trick to raising your Veterinary skill-- through afk macroing-- without actually having to do much of the boring stuff--- such as point and click.. wait for it.!?
Well, I hope this information helps. Have a great beginning to the New Year.
--Sturger--
