don't: tell your current employer
Of course, if you and your employer have both agreed that it is better for you to leave, you can say that you are looking for a new job.
If this is not the case, it's better not to say anything. Looking for work on the quiet is perfectly normal. You are responsible for your career. It's quite alright for you to take charge. In any event, you should never drop an occasional hint. We also recommend that you never tell a colleague in confidence. It's quite possible that your colleague might believe that your thoughts are already elsewhere. Which brings us to the following point.
do: continue to fully engage in your work for your current employer
People who have decided to leave are often not fully engaged and find it difficult to give 100 percent for their current employer. Nevertheless, it's important that you continue to perform well. You will leave with a good feeling and leave a professional impression behind you. And, of course, a positive reference will be important for your career later on. Applying for a job is an art form. You should never do it on the quiet during work hours. Find the appropriate time! For example, schedule a regular evening in the week to look through job vacancies or to write cover letters.
don't: make excuses so you can go to a job interview
First, recruiters are quite used to the fact that people are still working somewhere else. They can often accommodate your need, for example, by arranging an interview at the start or end of a workday. Are you allowed to take time out during your workday? Never lie to your colleagues or current employer. It's much better to be discreet. For example, say you need to leave early for personal reasons (that are not serious). Of, simply take a (half) day off.
don't: wear different clothes than usual
You naturally want to look good for your job interview. You only get one chance to make that first impression. What if you need to go directly to your interview at the end of your workday? Don't wear conspicuously different clothes. For example, don't wear a tie when you do not normally wear one. Don't wear a suit when you normally wear jeans and a shirt at work. Bring along a second set of clothes or put these in your car or in a sports bag.
don't: be negative about your current employer
What if they ask you about your reason for leaving in your job interview? Never be negative about your current employer. You could say that you've hit a barrier, for example, some processes don't work well or there are no growth opportunities for you. If you gossip about your employer, you leave a bad impression about yourself. And, will your potential boss wonder whether you will also say bad things about them in a couple of years?
do: change your LinkedIn, but be careful!
Do you use LinkedIn? If so, it's best to update your information. But, be careful. If you use the default settings, your connections will receive a message about your update. That can raise doubts with your current boss. So, first disable 'share profile changes' and only then update your profile.
You can also upload your CV onto job vacancy websites, such as Monster. Does your profile match a job vacancy? If so, it will be easy for recruiters to find you. It's best to make your CV anonymous just in case your current employer finds it by chance. So, you don't need to enter your name and you can describe your last job in general terms. Don't specify your current employer's company name, but that you work at 'a bank', 'a logistics company', and so on.
don't: search through all job vacancies
Ploughing through all of the current vacancies can cost you a lot of time. You already have a job and that probably keeps you busy. Have someone else do the footwork for you. For example, you can set a job alert so that relevant choices are sent to you. But, be careful. Getting a bunch of e-mail notifications can be a real pain, for example, if you receive such notifications during your work hours. So, it's best to always register with your private e-mail address, disable notifications, and check the recommended job vacancies at a regular time!