how do I use available alchemy points and teach my pet spells?

You just teach your pet the spells.

Let's say you have 6 Alchemy points. You can teach your pet three spells that cost 2 Alchemy points, or a spell that cost 4 points and a spell that cost 2, etc.

Try this thread for more info on spells and Alchemy points.

You can use the Alchemy points by Having a spell in you inventory, which are created In the sacred lands, then clicking on the scrolls, and use the 'teach spell to ___ ' option, and it will consume a certain amount of AP, which varies upon the scroll, they can be unequipped and you will receive your AP back.

More Info On Scrolls/Spells/AP Here