I guess as a mod, I'd love to say "Nothing". Hopefully not with maniacal laughter.
Practically, suspensions are a last resort and we'd really not have to suspend people. There's often some other stuff going on, a user's overall history and contributions.
While seemingly punitive, by design, suspensions are also meant to be cooling off periods - so a good way might be to actually show that you're aware of what happened, and display a desire to make it right. The first suspension's at most a week anyway.
In short a good first step if you're a new user is trying to work out how and why you went wrong. Unfamiliarity on its own isn't a good excuse for breaking the rules. Then you can open up a dialog with the mods, or simply wait it out.
If its really really bad, and you really need to get a CM to take a look, you can use the contact us link. That said, If you did manage to get the suspension reduced, please do remember the mods are the folks on the ground. If you do the same thing again - you will get suspended and longer.