How can I get the continuous-time transfer function coefficients (or poles and zeros) from the corresponding discrete-time TF and vice versa?
This is the formula that you need:
$$
H_d(z) = H_a \left( j\frac{2}{T} \tan(\omega_d T/2) \right),
$$
where $H_a(s)$ is the transfer function in the Laplace domain, and $\omega_d$ is the "warping frequency", i.e. the frequency where you want the transfer functions to line up exactly (usually chosen as the cutoff frequency or center frequency).