From AmE perspective:
We don't use "I'll go mad" much, more of an BrE expression, with one possible exception. I would definitely use the expression, "I thought I would go mad with grief"
go crazy = go insane = go nuts = go beserk
As to Artur's comment, that "go crazy" is less intensive than "go insane". I don't perceive that to be the case. I would "read" both expressions as being equivalent. Of course a lot depends on the speaker's emotional state of mind and sense of drama. :)
All of the above expressions could be used. It really depends on the circumstances: who is speaking (and to whom), how casual/formal the conversation is, itd
The expression most commonly used (and least ambiguous) would be to "go crazy"