It's unfortunate how nervous I have become to leave a bad work environment. You read all over LinkedIn that "job hopping" is a mortal sin, and will be held against you for the rest of your resume-writing life. This in turn causes us to feel awful for leaving somewhere, even if it's for viable reasons. But how do we get over that?
Firstly, we have to look at the situation from the eyes of an outsider. Let's say you work at a retail store, and heard from one of your coworkers that the assistant manager just put in their notice. What is your response to your coworker? To your assistant manager? In order to figure that out, put yourself back into the shoes of the person quitting. How would you want your coworkers talking about your departure? Would you want them to really talk about it at all? How awkward would it be to be told of your departure by someone you did not tell - maybe someone you didn't want to know? Kindergarten 101, here: treat others the way you want to be treated. Don't talk about another coworkers' departure without their knowledge, it doesn't make anyone feel good. Instead, you're your coworker that you don't want to talk about your assistant manager behind their back, as their departure is their business.
But back to your departure. Unfortunately, the situations from above can happen, depending on how gossip-inclined your coworkers are. All we can do to deter that is not be a part of it yourself when those situations occur. In order to keep that chatter down about yourself: tell no one other than your appropriate manager/human resources representative, and ask them to keep it to themselves when at all possible. Of course, I'm assuming at this point that you have a rapport with this person, so ask them in a way that you know they will respond well too.
Okay! So you've given your notice. Now you probably have some more shifts to work before that end date is here. Some people really take this as an opportunity to tone down their work ethic, and excuse it with "well, I'm leaving, so it doesn't matter." But then, that enables your coworkers to begin that gossip cycle formerly discussed. So keep your work levels and ethic at an even level; be sure to be tying up possible loose ends you may leave, and hand off any possible projects to appropriate parties. Just remain calm, and everyone else around you should as well. And always remember, if someone asks why, you are really not under any obligation to tell them anything. But don't lie. Come on now.
When your last day arrives, how you treat this situation is totally up to you. Personally, I never really let anyone know except for the coworkers that I'm closest too. Remember to still keep it professional, these people can all be used as references for you in the future (if they agree to it). It's never easy to be the new person, and it can be even harder to be the one who's leaving. Just take everything a day at a time.