Breeding itself has no skills. But if you want a pet character you need at the least:

Required
- Creature Control, every 2 levels is 1 pet point
- Advanced Creature Control, every 4 levels is 1 pet points
- Lores for the animal, helps with taming/dominating and every 4 levels is 1% off pet point cost
- Beast master, works in combination with pet level to unlock attacks
- Beast influence, something about engagement range/aggro, not very useful imo
- Taming or Dominating, taming allows you to trade/receipt the pets. With dominating, only you can dominate the animal if it loses loyalty

Optional
- Animal care, decreases time needed for pets to level up while out
- Herding, decreases pet point loss while controlling multiple pets
- Inspect, allows you to see points/stats of pets without having to own them, dependent on lores

How breeding works
For breeding, pretty much get whatever you want. Obviously you need a male and female, the offspring will get a clone of each stats from either parent, for instance if the males speed is 51 at lv25, the offsprings speed at lv25 will be 51 if the male's trait is passed on. And thats how breeding works, just mix and match the stats until you get an offspring you want, appearance works in the same way - more mixed between parents though. The fun stuff is the inbreeding chances that seem more dependent on appearance of mounts, stats and the amount of time the breeding session is going (last one being a personal theory).