$ZFC + \exists V_\alpha$ model of $ZFC \vdash Con(ZFC + \exists$ transitive standard model of $ZFC)$
and then
$ZFC + \exists$ transitive standard model of $ZFC \vdash Con(ZFC + \exists \omega-model$ of $ZFC)$
For the first one :
We can always find a countable extentional $M \subset V_\alpha$ elementary equivalent to $V_\alpha$. Let $M'$ be the mostowski collapse of $M$. $M' \approx M$ so $M'$ is model of ZFC. And because $M'$ is countable and transitive then $M' \in V_\alpha$ (since $H_{\omega_1} \subset V_{\omega_1}$ and $\alpha$ is surely far larger than $\omega_1$).
So $V_\alpha$ is the model of '$\exists$ a standard transitive model of ZFC'.
For the second one :
I don't really know how to do it... Does anyone have an idea ?