to get pissed off / ticked off / cheesed off (these are AmE, bear in mind that pissed means drunk in BrE)
to lose the plot
to fly into a rage
to throw a tanty (diminutive for tantrum so it's condescending)
to blow up
someone's blood started to boil
the red mist descended
Other than the first line, I think they're rather more British than American expressions.
In Aust/NZ English to "get shirty" means to get annoyed or angry, while "keep your shirt on" means "don't get angry", "calm down".
But to "lose one's shirt" means to lose all your money, usually through gambling.
pissed = drunk
pissed off = angry/annoyed
"spit the dummy"
"chuck a wobbly"
"chuck a spaz"
not a teacher