Recursive, Recursively Enumerable and None of the Above
You can choose the language of the halting problem

$\qquad \displaystyle B_1 = \{\langle T \rangle \mid T \text{ halts on } \langle T \rangle\} \in B$

and its complement

$\qquad \displaystyle C_1 = \overline{B_1} \in C$.

This is fairly standard material. The proof for $B_1$ not being recursive is the well-known diagonalization. Proving $B_1$ to be RE (recursively enumerable) is a tad tricky, involving interleaved simulation of multiple TMs, but is widely documented. If $C_1$ were RE, then $B_1$ being RE would imply that both are recursive; hence $C_1$ is not RE. This illustrates some of the techniques for such proofs in general.