5 ideas
A hotel with infinitely many rooms is full, yet a new guest can check in: every guest moves from room n to room n+1, freeing room 1. When an infinite busload arrives, each current guest moves from room n to room 2n, which empties every odd-numbered room for the newcomers.
Two sets are the same size if their elements can be paired off one-to-one with none left over, and counting is not required. Under this rule the natural numbers and the even numbers are equally large, because n pairs with 2n, even though the evens are a proper part of the naturals. For infinite sets the intuition that 'the whole is greater than the part' fails.
Cantor's diagonal argument shows that the real numbers cannot be listed in a sequence. Given any supposed complete list, you can build a new number that differs from the nth entry in its nth decimal digit, so that number appears nowhere on the list. The reals are therefore a strictly larger infinity than the naturals.
The inventor of chess asks the king for one grain of wheat on the first square, two on the second, and doubling on each square after that. The total is 2^64 − 1 grains, about 18 quintillion, which is more wheat than the world produces in many centuries. The request looks modest because people badly underestimate repeated doubling.
Mathematics is usually taught as finished answers and procedures, delivered before the student has felt the question that motivates them. That order kills curiosity and makes the subject seem dull and hostile. Starting from a puzzle or paradox that provokes a real question restores the play and pleasure that drive mathematical thinking.