|Don't ask to ask, just ask|
Here's some things that I've found useful (some of which have already been mentioned here):
1. Never be afraid to ask a question just because you're the new guy. Sometimes people have been in one environment for so long they take a lot of knowledge for granted, and once you ask the question, they generally are quick to realize this, and are happy to help you along.
2. Talk to everyone, but offer no opinions on political/enviromental thing even when they press you for one. They are testing you either for the company's benefit or their own. Either way, you don't want to bury yourself in a hole you hadn't realized was there yet.
3. OBSERVE, OBSERVE, OBSERVE. You will pick up a lot by watching the dynamics of the department (and the company). Again, DO NOT voice opinions on what you observe.
4. Find a way to infuse yourself in the organization. For example, my first IT job was as a Night Operator/Jr. Programmer. I quickly realized that I had an opportunity at night when I was waiting for jobs to finish to tackle jobs that the day time people didn't have time for. I chose to do the more high-profile ones whenever possible. This lead to my boss finding a way to move me to working days, and expanding my responsibilities as a programmer.
5. I can't help you with the keeping overtime to a minimum thing, as it has always been my philosophy to be as flexible as possible. This is my career, not my life, but those two items aren't necessarily unconnected (at least in my world).
Hope that helps in some way. Good Luck!
If you give a man a fish he will eat for a day.
If you teach a man to fish he will buy an ugly hat.
If you talk about fish to a starving man, you're a consultant.