Fantasy, Adventure, Satire
On first reading, A Connecticut Yankee in King Arthur's Court is fantasy, with the Yankee whisked away through unknown means to a far off land. It also carries shades of adventure, what with all the daring deeds, wrongs to be righted, and quests full of danger.
Most importantly though, Twain wants to make fun of these very genres he's employing… especially old Arthurian stories full of epic journeys and flowery deeds. Twain's engagement with the fantasy and adventure genres serves as a set up for the jokes Hank makes, pointing out how ridiculous this 6th-century world is. Which brings us to the third—and arguably dominant—genre this book falls under: satire.
We know we're in the realm of satire because pretty much everyone but Hank is endlessly being made fun of, either by Hank himself or by the scenarios they find themselves in. Here's Hank talking about the knights:
I had started a number of these people out—the bravest knights I could get—each sandwiched between bulletin-boards bearing one device or another. (16.2)
Do you see where he's mocking them? By saying he's employed "the bravest knights" to wander the kingdom wearing sandwich boards, the bravery typically associated with knights is mocked and, in doing so, as readers we are invited to engage with a more critical understanding of knights and their roles in society. This is satire in action, folks.