Hints:

For $\theta$, \begin{align*} p(\theta|\tau,\textbf{y}) &\propto \overbrace{\tau^{\frac{n}{2} - 1} \exp [-\frac{\tau(n-1)}{2} s^2}^\text{does not depend on $\theta$} -\frac{\tau n}{2}(\bar{y} - \theta)^2 ]\\ &\propto \exp[-\frac{\tau n}{2}(\bar{y} - \theta)^2 ] \end{align*}

For $\tau$, \begin{align*} p(\tau|\theta,\textbf{y}) &\propto p(\theta,\tau|\textbf{y})\\ &\propto \tau^{-1} \underbrace{\tau^{n/2}\exp[\frac{-\tau}{2} \sum^{n}_{i=1}(y_i - \theta)^2]}_{L(\theta,\tau;\textbf{y})} \end{align*}

Hints:

For $\theta$, \begin{align*} p(\theta|\tau,\textbf{y}) &\propto \overbrace{\tau^{\frac{n}{2} - 1} \exp [-\frac{\tau(n-1)}{2} s^2}^\text{does not depend on $\theta$} -\frac{\tau n}{2}(\bar{y} - \theta)^2 ]\\ &\propto \exp[-\frac{\tau n}{2}(\bar{y} - \theta)^2 ] \end{align*} and spot the Normal density in $\theta$

For $\tau$, \begin{align*} p(\tau|\theta,\textbf{y}) &\propto p(\theta,\tau|\textbf{y})\\ &\propto \tau^{-1} \underbrace{\tau^{n/2}\exp[\frac{-\tau}{2} \sum^{n}_{i=1}(y_i - \theta)^2]}_{L(\theta,\tau;\textbf{y})} \end{align*} and spot the Gamma density in $\tau$.