One thing I often get questions about from friends and coworkers is how to gracefully exit their positions (or handle a position that is falling apart, ie leaving on "bad terms"). So here's some thoughts on the termination process.
How much notice do I give?
As much or as little as you feel comfortable with. Employers like to tout the "2 weeks notice" as the defacto standard but frankly the majority of states have this whole legal policy that says an employer can terminate you at any time for no reason whatsoever. So frankly if employers are willing to toss you on your ear with no notice then I don't really see the logic behind feeling compelled to give two weeks notice. So yes, if you're changing jobs and your new employer wants to start you early, do not feel bad about giving short notice because it's not them you're trying to please anymore but your new employer. That being said however, you really don't want to burn your bridges so if it doesn't cause hardship go ahead and give that 2 weeks notice.
In addition, especially in IT, be prepared to be terminated the moment you give notice. Many companies for liability purposes have policy that says as soon as you give notice they need to disable your accounts and escort you to the door. They may or may not give you severance pay up to your notice, though it is known to happen and in my opinion is the classy thing to do. In other words, do not give notice unless you are comfortable losing your job and income that day. If you've accepted another offer pending anything (background check, etc) then do not give notice until all that stuff clears.
Don't burn your bridges!
More often than not in the business world success depends upon your contacts. When you leave an organization make sure to be civil and get contact information from anyone who is skilled and/or you got along with. You never know when you might be able to help someone find a job in your new organization or when they might come across an opportunity that suits you. In addition, if you're leaving and disgruntled, do refrain from having a bitch-fest at your exit interview. As much as you'd like to tell the HR rep what an asshole your boss was, or how your coworkers made your life miserable, it does absolutely nothing for you. You're leaving, it's no longer your problem or your battle to fight. Let nature take its course... if people in your position keep quitting eventually upper management will put two and two together.
Accepting a counter-offer
Many career advice types will tell you never to accept a counter offer. Being that you've already announced your intentions to quit, even if you accept a counter offer there is a legitimate fear that it will be held against you sometime in the future. Do you blame them? You've shown yourself at this point to be "disloyal" since you were out job hunting. If you are quitting because you are getting an advancement opportunity or more salary elsewhere a better way to handle the situation running up to the resignation is to schedule a frank talk with your manager. Tell them that you really want a promotion path or that you do not feel like you are being compensated at your market value. Be prepare to back up these statements by highliting your accomplishments and cost savings you bring to the organization. If they are unreceptive or make promises and don't keep them then it is time to move on.
What to do when things are going poorly
Is someone "out to get you"? Do you feel like you are going to be terminated? Probably the best thing you can do in this situation is to start documenting. You should know that it is difficult to collect unemployment compensation if you were terminated for cause, so if you expect to be out of work for a while your best defense is to start documenting that your termination was not for cause. This includes printing out hard copies of email communications, copies of your performance reviews, and any other information. You'll want to document both the good and the bad. Performance reviews are particularily helpful since if you can show that you got high marks and then were suddenly terminated for an unjust cause you can use it as evidence. This is particularily helpful in harassment and discrimination coverups. A friend of mine once was sexually harassed by a boss and when she complained got the run around and was eventually fired. Because she had documented evidence of outstanding performance for her last few years of service she was able to litigate and win a wrongful termination lawsuit.
Also, try to examine why things are going so poorly. Are you too blunt? Do you make people uncomfortable? If you're on the way out there's no reason why you shouldn't go around and ask for some information from coworkers on how you are perceived and what they think you can do better. Sure it's a pride pill to swallow, but if you can take those lessons into your next position you'll be a better person for it!