Classes
In the third year, students were able to select from a more diverse list of courses. As a result, not all students shared the same curriculum.
For example, Harry added only Divination and Care of Magical Creatures from the previous year.
Most important of all, he had to buy his new schoolbooks, which would
include those for his two new subjects, Care of Magical Creatures and
Divination.
Hermione, on the other hand, had quite a few more:
"Well, I'm taking more new subjects than you, aren't I?' said
Hermione. "Those are my books for Arithmancy, Care of Magical
Creatures, Divination, the Study of Ancient Runes, Muggle Studies --"
The courses offered were (elective courses in bold):
Not all of these classes involved spells or potions.
Potions
"Everyone gather 'round," said Snape, his black eyes glittering, and
watch what happens to Longbottom's toad. If he has managed to produce
a Shrinking Solution, it will shrink to a tadpole. If, as I don't
doubt, he has done it wrong, his toad is likely to be poisoned."
They had Potions that afternoon, which was an unqualified disaster.
Try as Harry might, he couldn't get his Confusing Concoction to
thicken, and Snape, standing watch with an air of vindictive pleasure,
scribbled something that looked suspiciously like a zero onto his
notes before moving away.
The function of a Confusing Concoction may be explained in the fifth book, if it is the same as a Confusing Draft:
These plantes are moste efficacious in the inflaming of the braine,
and are therefore much used in Confusing and Befuddlement Draughts,
where the wizard is desirous of producing hot-headedness and
recklessness ...
"How's she doing it?" Ron muttered to Harry one evening as Harry sat
finishing a nasty essay on Undetectable Poisons for Snape. Harry looked
up. Hermione was barely visible behind a tottering pile of books.
Spells
The main textbook is the Standard Book of Spells, Grade 3.
According to Potterwiki
This is the first book in The Standard Book of Spells series whose
contents will not be viewable on Pottermore.
Also according to the wiki, the spells for the third year can be found in Harry Potter and the Prisoner of Azkaban (video game), but include the Patronus Charm. So they should perhaps be taken with a grain of salt.
In any case, the spells are:
Disarming Charm
Draconifors Spell
Apparently this transfigures objects into dragons.
Freezing Spell
Lapifors Spell
Apparently this transfigures objects into rabbits.
Lumos Duo Charm
Mending Charm
Seize and Pull Charm
Snufflifors Spell
Apparently this transfigures objects into mice.