The Hollingdale biography of Nietzsche would be a good place to start. If you just want to get immediately into his writings read them in this order.
You can skip Birth of Tragedy, Wagner, and Twilight of the Idols if you want since they aren't crucial to his thought and start immediately with:
1.The Gay Science
2.Beyond Good and Evil
3.Genealogy of Morals
4.Ecce Homo
5.Zarathustra
6.Antichrist
If you are only going to read one book by him I recommend The Gay Science.
>>3 Objectivism didn't exist when Nietzsche was around nor did existentialism.