Like the others said, you should say "Don't get mad at me" or "Don't be angry with me". Using "at" means that the anger is being directed towards you, and "with" would be used in the same way in certain sentences. Some examples: He is happy WITH me. She is upset WITH me. I am angry AT them. I am mad AT him. You can see that the emotions that are most direct and harsh (mad, angry) use the word "at". The less harsh emotions use "with". However, the use of these words can change depending on the context, but "on" and "to" are not really used in sentences like those.

I would say "Don't get mad AT me." Or, "Don't get ANGRY WITH me."

Also, you could say "Don´t get mad at me"

Only "Don´t get mad with me" is correct.