Let $G(V, E)$ be a directed graph with edges of either blue or red color. And let $u, v \in V$ and $u \neq v$. How do I find a path from $u$ to $v$ such that all blue edges appear after all red edges (in other words, the path consists of only red edges at first as it leaves $u$ and consists of only blue edges as it approaches $v$). The path is allowed to contain only red or only blue edges.
What I have in mind is to run a breath-first search (BFS) from $u$ on only red edges until $v$ is found. If $v$ is not found, perform BFS from every vertex that we have visited before but this time searching only on blue edges until $v$ is found, skipping any vertices we have visited previously. But is this the best way?
Find the set $W$ of vertices $w$ reachable from $u$ via only red edges. BFS on only the set of red edges of $G$ will do.
Find the set $X$ of vertices $x$ where $v$ is reachable from $x$ via only blue edges. BFS on only the set of blue edges of $G$ with the directions of the edges reversed that is, will do.
If $W$ and $X$ intersect at a vertex $y \not \in \{u,v\}$ then letting $P_{uy}$ be a dipath in $G$ from $u$ to $y$ of only red edges and $P_{yv}$ be a dipath in $G$ from $y$ to $v$ of only blue edges, then $P_{uy}P_{yv}$ suffices.
If $W$ and $X$ do not intersect at all then there is no such path.
If $W$ and $X$ intersect at $v$ there is a path of all red edges from $u$ to $v$.
If $W$ and $X$ intersect at $u$ there is a path of all blue edges from $u$ to $v$.
If $W$ and $X$ do not intersect outside of $\{ u,v \}$ then there is no path as described in the OP that uses both red and blue edges.