Hmmm... this is a big question. First thing to do - sign up for a track class, whatever you can afford. It will allow you to greatly increase your enjoyment and handling of the car.
Short of that, I have to ask - is this your first time using a manual transmission?
If so, normal launching is a case of coordination. You want to let the clutch out to the friction point, or the point where it starts to engage the motor. You give the motor a bit of gas at the same time. Practice starting over and over, and you'll get smoother with the gas/clutch combo. If you really must abuse the car and go for the fastest launch, you do that by pushing the clutch in, reving the car up to a few thousand RPM, and let the clutch out suddenly (otherwise known as dumping the clutch). Done right, the car will spin tires and accelerate quickly, but it also may not go straight. I wouldn't even recommend doing this until you have much more experience.
Speed shifting is highly overrated. You actually want to work with the transmission and the motor. If you're accelerating hard, you will shift at a higher RPM because the motor generates more power the higher the RPM; if you're taking it easy, you will shift at a lower RPM (maybe 3K) because it stresses the motor less and increases fuel economy.
Now if you're shifting from 2nd to 3rd (an upshift), the motor is going from higher RPM to lower, right? That's the whole point of an upshift. That means, when shifting, that you usually want to give the motor a bit of time to drop RPMs before going into the new gear. In other words, put the clutch in, pull the stick out of second, wait a *tiny* moment, then put the stick into third and let the clutch out. That pause is greater at low RPM, just because it takes the revs longer to drop, and you get closer to a full speedshift if your up near redline (hard acceleration). If you get the pause just right, you'll feel that it goes into the next gear easily.
On downshifts, you're going from a low RPM to a high RPM. This is why the Z has SyncroRevMatch, which I suggest you just leave on. Basically, you downshift from 3rd to 2nd, and the computer will automatically match the motor revs from one gear to the next. Leave this be.
Anyway, all of this is better done with someone there who can explain it. Don't be afraid to learn - people are usually happy to teach. Just don't go driving around like a jackass. This car may not be faster than a Z06, but it's very, very fast, way faster than a 1980's ferrari 308. You need to learn how to use it and respect it, which is where driving school comes in.
Bob Bondurant is one of the best driving schools in the country, and he's right in Phoenix:
Bondurant.com :: High Performance Driving School
They offer a range of classes, from a highly worthwhile half-day class to the pricier (but totally awesome) full track weekends. If it looks expensive, think of what you spent on the car and realize that spending a tiny fraction of that to be able to enjoy the car (not to mention not wreck it) is totally worth it. I promise you will not do a class with them and not be glad you did it (and no, I don't work for them, or know anyone who does).
Anyone else got advice?