Let's look at the form of the allowed paths. As a regular expression, it looks like:
$$r^*\ |\ r(r|g)^*\ |\ rr^*g(r|g|b)^*$$
We can test the three cases separately and then choose the shortest path found among the three cases.
The first case is easy: just remove all the green and blue edges and then find the shortest path from $s$ to $t$ as normal. Of course, there may not be a path from $s$ to $t$ in this red-only subgraph; if this happens, we can say the distance for this case is $\infty$.
For the second case, start with the original graph and remove all the blue edges, leaving only red and green edges. Then add a new vertex $s^\prime$, which has a directed edge to each vertex $v$ exactly when there is a red edge from $s$ to $v$ in the original graph. Then check for the shortest path from $s^\prime$ to $t$ in this modified graph; this corresponds exactly to the shortest path from $s$ to $t$ in the original graph that has the form $r(r|g)^*$. Again, if no such path exists, we can return $\infty$ from this case.
See if you can figure out the third case now. 🙂
I have asked and answered a more general form of this question here.