Some more theorems that seem obvious but require non-trivial proofs:

https://en.wikipedia.org/wiki/Intermediate_value_theorem

https://en.wikipedia.org/wiki/Jordan_curve_theorem

https://en.wikipedia.org/wiki/Isoperimetric_inequality

https://en.wikipedia.org/wiki/Hairy_ball_theorem

https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstei...