" Animals are such agreeable friends -
they ask no questions, they pass no criticisms."

– George Eliot

This skill is essential for a tamer and anyone who wishes to ride a firesteed, a mare, a unicorn or a frenzied ostard. The level of your animal lore will determine how well or not your pet listens to you and opens up the menu with your pet's stats as well as tells you what mood he is in. This skill is affected by the anti-macro code, so you will need to move around to gain skill.

You can go with the slow method, using animal lore directly on any animal you see. However, you will need to lore a large number of various animals as you can't gain off the same one over and over again. You also gain animal lore through taming and veterinary.  This is a very long process, but like in all skills, if you're willing to take the long road, it's always quite rewarding.

And then, there is a faster way.

Tame a bear and a horse. Ride your horse but make your bear stay. Mark a rune right next to your bear. Recall to a boat and open a gate to the rune you just marked (you can't cast gate on a boat key) go through the gate, get the bear to follow you then get back on the boat.  Release the bear.  Start moving slow forward heading north-south.  Get your horse to attack the bear and heal the horse.  You will gain a lot of veterinary. As soon as you get an animal lore gain, stop the boat.  Lock veterinary and keep healing the horse.  You should gain every time.  When you stop gaining, try a couple more times at the same spot before moving.  If no gains, unlock veterinary and start sailing again until you find another gain tile.  I found I got the best animal lore gains when veterinary was between 54 and 61.  Once you go over 70, drop veterinary to low 50s and start again.