We developed a theory of electric and thermoelectric conductivity of lightly doped SrTiO$_3$ in the non-degenerate region $k_B T \geq E_F$, assuming that the major source of electron scattering is their interaction with soft transverse optical phonons present due to proximity to ferroelectric transition. We have used kinetic equation approach within relaxation-time approximation and we have determined energy-dependent transport relaxation time $\tau(E)$ by the iterative procedure. Using electron effective mass $m$ and electron-transverse phonon coupling constant $\lambda$ as two fitting  parameters, we are able to describe quantitatively a large set of the measured temperature dependences of resistivity $R(T)$ and Seebeck coefficient $\mathcal{S}(T)$ for a broad  range of electron densities studied experimentally in recent papers.  In addition, we calculated Nernst ratio $\nu=N/B$ in the linear approximation over weak magnetic field in the same temperature range.