It is pretty easy to level any pet in vanilla Classic WoW and get loyalty.
First of all, if you get a pet that is level 59 and you want to level it to 60, you can just go and kill some mobs. Pick an easy area with lvl 50 mobs like Burning Steppes.
Be sure to keep your pet happy (green icon), have enough food.
Level the pet to 60. By the time he has reached 60, it should have level 2 loyalty.
To gain level 60 from 59 for your pet, it should take about 2-3 hours of killing mobs.
After that, you can just sit anywhere with the pet on and as time will pass, he will automatically gain loyalty.
Leveling a pet from level 30 to level 60 for example will be a chore. Rather get a pet that is high level, at least 57, 58 or 59 if you are a level 60 hunter.