path problem.... please give me some explanations

I am now trying to solve the path problem but I dont know how to start for it...
I need some explanation on that problem and if possible could you tell me how to implement it in C++.
I checked out the net for Dijktra alg, but I am still donot understand much. If the distances to goal are all the same, just like the bitmap I include below when the agent goes from I or D or G to the final goal H. This time I will randomize the way for my agent to get there, is it right ?
One more inportantthing I would like to ask is that when my agent start from A, I will randomize to choose a way is it right ?
But If I choose a way that is not the most optimized one like the point B, how will I correct it ? Could you help me with this ?
Please give me some explanation...

Thank you very much...
Regards,
Nina