On the motherboard, you have these connections (with both + and -, but we'll ignore those for now):
Power Switch
Power LED
HDD Activity LED
Reset Switch
On your case, you have these connections
Power Switch
Power LED
HDD Activity LED
Reset Switch
Is this correct? These are standard on most cases and motherboards. On some cases there may be additional LED's. To power on the computer you must short out the Power Switch +/- pins on the motherboard. That is the purpose of the power switch. If you want to test if you can turn on your PC, take a small metal object (flathead screwdriver, paper clip, staple, etc) and touch it between Power Switch + and - pins. This should turn on the PC (I've done this numerous times on many different boards, helps when you don't know what pins do what).
I didn't use a grounding strap when I built any of my systems, never had any issues. If your PSU is mounted and plugged in, you can simply touch any bare metal on the case to ground yourself (as the whole case should be grounded when the system is plugged in). None of the voltages outside the PSU are strong enough to shock you.
Edit:
On some cases (my roommate's NZXT and apparently yours as well) has an LED in the Reset button. I'm not sure what line this would be connected to, probably just Reset Switch. You could confirm this by pressing the switch while the LED is lit. If the LED goes out then they are connected together, if connected to Power Switch pins on the motherboard the PC should also turn on. Note however that the Power LED and Power Switch are always separate (else the Power LED would glow even when the PC was off).