Some people view their hairdressers as therapists. Like shrinks, hairdressers will let you ramble on incessantly about your life, thus, making you feel better about yourself. No wonder it can be so hard to leave an old hairdresser for a new one. Maybe your hairdresser has cut your hair too short, has not styled your hair as you like it, and you want to find a different hairdresser. This is what you could do when you really want to leave the one you have now and try someone new.

[edit] Steps

Give your stylist a second chance. Be specific about what you want done at the beginning of the appointment. If you still don't like it, explain to your stylist what it is that you want and what she/he is not doing. If you want a certain look bring pictures from a magazine. Most hair stylists are cool about adjusting their work to meet customer specifications. If the stylist acts like a total prima donna about it, then she/he is making it all the more easy for you to leave.
Don't let yourself feel guilty. It is, after all, your money and your hair. The beauty industry is about making the customer feel better about his/herself.
Trust your own judgment. You're the one who has to look at yourself in the mirror everyday. If the advice your hairdresser gives you doesn't work, don't let him/her change your mind.
Get a second opinion. See what another hairdresser thinks of your hair. Some of them will consult with you.
Realize that you don't even have to tell them you are leaving. You could just start going to another salon and be polite to them if you ever see each other again in public.
Be honest. If you are friends with the person, let them know what you like about them, but that you think it's better to go with someone else.

[edit] Tips

Watch the other stylists in the salon you are using. You might find one that you like and would like to do your hair.
You can always make an appointment at the same salon with another hairdresser.
It is easier to do this than to just find another salon, though it may be a little awkward. If you don't find anyone else there that you prefer, then by all means just do not make another appointment at this salon.
A note from the U.K: Hairdressers are there to provide a service, if you are unhappy then go to another hairdresser or salon: never feel guilty for moving on, change is good!

[edit] Warnings

When telling your old stylist you're switching, (if you decide to tell the stylist) be as polite and respectful as possible! You don't want to make him/her feel bad.
Never complain about the hairstylists work!