Shortest travelling cost if we have bunch of points in 2D plane
This is the Traveling Salesman Problem with the $L_1$ metric (also known as the Manhattan distance).

Unfortunately, this version of the TSP is also NP-hard.  Therefore, there is no algorithm that always produces the optimal solution, is efficient, and works for all inputs.  Instead, you'll have to use a heuristic or approximation, or accept a solution that can take exponential time in the worst case.  Dijkstra's algorithm won't suffice.

Strictly speaking, the standard formulation of the TSP doesn't allow you to visit any point more than once.  However, in this situation allowing you to visit a point more than once doesn't make the problem any easier, since visiting a point more than once will never help you find a shorter solution.

Perhaps the following paper will be useful:

The L1 Traveling Salesman Problem.  Donald C.S. Allison, M.T. Noga, Information Processing Letters, vol 18, 1984, pp.195-199.