Coming from a 23 year old sysadmin out of college working for a media company on the web side... I started a couple of years ago in your same situation, so here's a few tips that's helped me.
1) Be personable, think positive, smile and laugh.
2) Be ambitious!! I wouldn't go as far as turning on the emergency fire suppressant to see how it works, but if you think you have an idea to better the company, don't be shy.
2) Be proud of your mistakes, and react appropriately. The first thing I do if I cause a catastrophe is fess up and fix it. If I can't fix it, the higher up's or google will. There's usually a laugh in the end, if there isn't, you're probably working for the wrong company.
3) Understand your role and people will offer their help.
After college, the day I got my job, I was ready to start studying for certification. A couple of months past and I quickly realized that I couldn't study work and exams at the same, so I didn't worry about certification. It's been 2 years now, and I actually just signed up for my first MCSE exam to write next month. Work should be your focus.
Organization is, and still is a fine art to me. It's been a running joke in the department because I'm so horrible at it. Yes, I'm horribly organized and I'm able to make a joke out of it. I first started with several post-it notes, not a good idea. I graduated to little paper flip books, not too bad, not too good. The big winner is the portable white-board on my desk, in front of my keyboard between my monitor, combined with outlook calendar and preferred ticketing system.
Daily visits to serverfault is also a must.
And don't listen to Zenham, those old guys don't know what they're talking about ;).