What are the new spells and where can I find them

Look in your menu->magic->active. Hover over each item, and you will see in the description of the effect what gave you that, and it will say 'blessing of talos'.

I have done a lot of testing on this and have yet to find a way for a companion to actually learn a spell. If you give them staves that have spells on them they will spam them. As long as they are not conjuration spells. I used Lydia and when given conjuration staves she will revert to her bow and will not even equip the stave that summons. I am also unable to unequip her bow and arrows. They seem to be "Basic" companion items they always have.

So in short: No they cannot learn spells.

Although: If your companion is a spell caster they will cast the spells they already had but will not learn more.