Training

Dan

I'm new to animal training and I'd like to get my pets HP as high as possible. After preliminary training what should I select to get its HP up?

Bilbo

copy your tamer over to TC and make your mistakes there.  Do this as many times as needed till you get the pet build that you like then go back to prodo shard and have fun.

Yadd_of_Legends

On the first round, you probably want to start with getting the resists up to minimize damage to the pet. 
https://stratics.com/threads/pet-training-a-z.392869/
You can also check out some of the other guides in the tamer forum
https://stratics.com/forums/uo-tamer.419/

Bruce

HP is point based in the taming system.

What kind of critter are you working with?

Even the same species will have some variation in stats, so if you look before you tame, sometimes you can find one with higher HP than average.