I've done a fair amount of troubleshooting on Windows machines and here's some guidelines I've distilled when you come across a problem.
Before contacting support, try troubleshooting yourself. You'll become a better person and world peace will be just around the corner, trust me on that.
Make sure the situation you are experiencing is actually a problem. This is crucial, it really is. A message popping up sometimes is just informative, not a real error. See also 6.
Try the same action and see if you get the same result. This will prevent the embarrassing IT support effect from happening where it always works when someone from IT tries the same action.
Always reboot. Seriously, this solves 50% of all problems.
If possible, try it on another computer, or use a different browser.
Consult the help file, every now and then it contains useful information.
Read error messages, every now and then they contain useful information that might actually help you solve the problem. I know this is really hard to believe, but do try.
Google remains your friend.
Ask a colleague, you'll be surprised how helpful they can be.
Reboot again, just to be sure.
Use the preferred way of contacting support, be that an on-line portal, e-mail or telephone. The benefits are obvious: your case will be logged, creating a trail that can be useful in the future. Multiple people will have access to it, making it hard to be missed. The system will also provide managers with reports that can help them in preventing certain errors from happening, creating clearer documentation (haha) and so on.
Be nice. As a helpdesk engineer I've had to put with a lot of abuse over the phone. Being rude or shouting will never get your problem resolved faster, on the contrary. The same applies for written requests. Don't use caps as this is the equivalent of shouting (IT REALLY IS).
Clearly state what you are trying to do, what result you expect and what result you actually get. I cannot stress this enough, people often assume they know what the solution to their problem is and just ask you to perform that action, without mentioning their original problem.
Specify the severity of your problem. Is it only you that is blocked, an entire group, is there a workaround, ...
State what you have already tried to solve the problem, like rebooting 3 times (it always seems to be 3). Most of the times support will reply with the standard troubleshooting tips. Saves everyone time when you explain what you already tried.
Be honest. I've wasted many hours trying to identify root causes because users were afraid to admit they had done something stupid. Everyone makes mistakes, so fess up. And do be honest about having rebooted, support can sniff out if you're lying about this.
Send all material that may help support. This can be screenshots, files that are giving problems, error reports, ... A screenshot is worth more than a thousand words. I've been able to solve cases in 10 seconds based on a single screenshot, where 20 previous e-mails only led to mounting frustration. If you send a screenshot of a browser with toolbars installed, be ready to take some flak though.