There can be a number of reasons you need a new person to teach you on your chosen instrument. It may be that, after a few lessons, you feel they're not right for your personality. Or it could be that they've taken you as far as they can, and you need someone else who can push you harder. It could even be because of a change in your schedule or something as simple as that.
How to Tell Your Old TeacherEven if you discover the teacher isn't all you'd hoped, be polite. Explain that you feel this isn't working out, and you feel the need to look elsewhere for tuition. Any good teacher should empathise, and understand, maybe even more than you do, about the relationship involved.
If you feel you need a teacher to take you to a higher level, explain that carefully. Most teachers will be aware of their limitations, and will probably have contacts who might serve you well. It's quite common to change teachers for this reason as you grow, and many specialise in different levels of players.
It becomes harder if you need to change tutors for your child. First of all you have to know they're dissatisfied or need to learn on a higher level, which comes down to you being very involved in their musical education. A reluctance to go to lessons is always an indication of something - you're the one who has to discover what, exactly.
Finding a New TeacherIf you need a new teacher, you're not back to square one. You have your experiences to draw on now, a much clearer idea of what works and what, exactly, you need. If you seek a more advanced teacher, playing your best piece for them and ask for a critique. Find out exactly what they can offer you, how they can improve your playing.
Should you be seeking a new teacher because of personality, then this time you come in with some knowledge, however, basic, of the instrument. That makes it easier to get an idea of how a new teacher can work with you. If it's for your child, sit and watch how the two interact over a few minutes of teaching. A happy child will be eager to learn, and when young that's as important as what they learn.
A New RoutineYou should remember that, although you're the paying customer, you're also the pupil. Each teacher has their own way, and will probably expect slightly different things from you - but that's all to the good. A shake up in routine can energise you.
Allow two or three sessions to be sure things work well between you and your new teacher. But one thing you don't want is to be constantly changing teachers - you won't learn that way. With your experience you should have been able to find someone good, so, unless an unforeseen problem occurs, stay with it.
It can be very gratifying when you switch to a new teacher and you seem to progress by leaps and bounds. However, don't constantly expect that. There can often be a big jump at first, but it will level out after a while. Don't be disappointed; it's simply the nature of things. Above all, don't give up.