Is every set of measure zero countable? I know it is true that every countable set has measure zero, but is the converse true. Is it true that every set of measure zero is countable?

No, and it's not necessary to jump to the Cantor set (which requires some intense mathematics) in order to see this. Think of a line (such as $\Bbb R \times \{0\}$) in $\Bbb R^2$ - it is a null set, but clearly not countable.
In general, if you have at least a vague intuitive idea of the concept of "dimension", everything that is of dimension $\lneq n$ is a null set in $\Bbb R^n$ (with respect to the usual topological and measurable structures) - and most of these subsets are clearly not countable.
