Your problem is non-trivial because e.g. if you greedily go as far as you can up or to the the right, then you might encounter a tight maze of objects that requires a crazy zig-zag path to finish, whereas if you stopped before the tight maze you might be able to change directions fewer times by essentially going around the maze. And you could encounter this dilemma anywhere along your path, not just in the beginning. One way to solve this problem is to use Dijkstra and define a grid of locations you can travel to, and then define a move to be 2 steps long instead of 1 step long. Define the distance between two connected grid points to be very small if the move is pure horizontal or pure vertical in one oriented direction, and very large if the move changes direction in the middle. Then, assuming the path length from start to finish is even, the shortest path from the start to the finish in this double-move framework will be the path that minimizes the amount of zig-zag. If the path length from start to finish is odd, then use the grid points one space away horizontally and vertically to start from and then the path length from start to finish will be even (although you'll have to run the path finding for both possible modified starting positions).
