What are some examples of the different kinds of irony in Candide?

As one observes with Candide, irony is a weapon in Voltaire's hands. He uses it to attack the irrationalities and abuses of the world.
Possibly one of the most striking examples of situational irony in Candide lies in the character of Pangloss, introduced as Candide's teacher and described as "the oracle of the family." With his role and reputation, one might think him wise or knowledgeable, but when we actually hear what he teaches, we find that it's all nonsense. Consider his words from Candide's opening chapter:

He proved admirably that there is no effect without a cause, and that, in this best of all possible worlds, the Baron's castle was the most magnificent of castles, and his lady the best of all possible Baronesses.
"It is demonstrable," said he, "that things cannot be otherwise than as they are . . . Observe, that the nose has been formed to bear spectacles—thus we have spectacles. Legs are visibly designed for stockings—and we have stockings."

The nonsense goes on; it's all meaningless and banal. Note that the characters themselves continue to treat Pangloss as a font of wisdom and knowledge—as the greatest philosopher of his age—adding a further touch of dramatic irony to this situation. They believe in his reputation for wisdom, even as the audience knows him for the fool he is. Moreover, as if this alone weren't enough, Pangloss's advice can be actively destructiv. For example, in chapter 5, Pangloss actively convinces Candide not to rescue the Anabaptist (who had previously aided them) from drowning, trying to rationally demonstrate that the Anabaptist is meant to die and that it is best that Candide not intervene to save him.
Voltaire makes effective use of verbal irony as well. His narrator can be deeply sarcastic. In Candide's third chapter, the narrator describes in gruesome detail the devastation unleashed by war, with the intense and monstrous suffering involved, but he accompanies this grisly scene with exalted language, opening the chapter with the telling line, "there was never anything so gallant, so spruce, so brilliant, and so well disposed as in the two armies." In these ironic tensions between what the narrator tells us and what he shows us, we find a powerful indictment of warfare and the suffering it causes.
As another example of verbal irony, I would point out the title of chapter 6: "How the Portuguese made a Beautiful Auto-da-fé, to prevent any further Earthquakes: and how Candide was publicly whipped." This auto-da-fé is described in the following terms within the text itself: "it had been decided by the University of Coimbra, that the burning of a few people alive by a slow fire, and with great ceremony, is an infallible secret to hinder the earth from quaking." As you can imagine, there is nothing beautiful about such a scene, even if Candide's narrator presents it in those terms. Furthermore, in a final touch of situational irony, Voltaire adds that on "the same day the earth sustained a most violent concussion." In other words, this ceremony did not even work.
This answer has only drawn on a few examples and has focused only on Candide's opening chapters. I'd suggest you continue reading to try to find more examples of the various forms of irony.