Monster Classes are different and aren't available from the NPC at Alltrades Abbey, by default. In order to switch to a Monster Vocation, you need to have an item called a Heart. Each monster that drops a Heart (not every single monster in the game does) drops it at a pretty low percentage out in the wild. Once you have found one, you need to have it in that character's inventory and then speak to the NPC at Alltrades Abbey to be able to switch to that Monster Vocation.
There is one slight catch to increase the chances of a Monster Heart to drop. After you get the Monster Munchies from Monster Meadows (via the portal near the plinth in The Haven), you will notice that monsters will come up to you after battles and you can send them to Monster Meadows. You can also win Hearts from some of the in-game casinos. Unfortunately, you don't unlock the actual use of Monster Meadows until later in the game, which is the ability to create Traveller's Tablets with the monsters you have tamed.
You can send monsters to hunt in a tablet that you create yourself, with the ability to send three inside of one. Once you do this, a Traveller's Tablet will be created and you can go in exploring yourself, with those monsters you sent in composing the majority of the enemies you find. It is with this method that monsters have a higher chance of dropping their Hearts, with an even higher chance if the monster is the boss of the tablet. Note that the Hearts are consumed when you have a character change into that class, so if you wanted all of your characters to change into a Slime, you'll need a total of FIVE hearts.
Important: Even with the above method, some monsters that have Hearts do not drop them and must be acquired through other means.
Important: The battles listed on the following pages to advance to the next level in a Vocation is a rough estimate and not always set in stone, as there is some kind of hidden counter that may advance it faster.