Jack,
Welcome to the forum.
I did change all my fluids out before my first DE and still think that is prudent. You will get lots of opinions on fluids here (or anywhere). If you are an aggressive driver, you can easily see oil temps of 300 degrees, get a tranny high temp warning which comes in at about 276, and who knows how hot the diff is, probably hotter than the tranny. Even good synthetics don't hold up forever at these temps. This is why most track hounds eventually get an oil cooler and many get a tranny cooler (I have both). You certainly don't need them for your first DE though. Go have fun out there.
I chose to run these (all of which meet GM specifications for the Z06):
AMSOIL Series 2000 0w30
AMSOIL ATF
AMSOIL Severe Gear Extreme Pressure Synthetic 75w90
This comes in handy:
AMSOIL Hand Pump
Note: tranny will take about 3.75 quarts, and diff will take about 1.75 quarts.
AMSOIL Ordering Information
AMSOIL Preferred Customer Program
For brake fluid I use Wilwood EXP 600 Plus which is comparible to Motul RBF 600 in cost and performance, but is sold by an American company. Note: this fluid isn't a DOT 3 fluid, but I'm guessing your car isn't under warranty anymore anyway.
