That's not the traveling salesman problem at all, it's just shortest path and is not intractable like the TSP. The Dijkstra Algorithm will give you the perfect solution and as far as I know is the fastest as well. That is actually what google does every time you ask for driving directions - it treats intersections as graph nodes and streets as graph edges, weighs edges based on length and average speed limit, and solves for the shortest weighted path.

Traveling salesman seems like a trivial modification of this problem, but it doesn't have any "tricks" to it like the Dijkstra algorithm. You have to generate all possible routes (brute force) and sort them, which eventually turns into a LOT of work, kind of like guessing a secure password.