There are many good sources of well-trained music teachers in New Jersey -- and also a few to avoid. For string instrument lessons, an excellent teacher is Professor Anthony Serafini of Centenary College, who takes private music studenst as well as college ones. An excellent piano teacher is Irene Frail, who can be contaced via her husband Prof. Robert Frail also at Centenary College. Another good source of piano lessons is Ridge & Valley Music Together, also in Hackettstown. Also noteworthy are  Kristine Winkler who offers voice lessons in Englewood, and Rebecca Ruivenkamp who also offers singing lessons in Jersey City. NJ. If your interest is in guitar (several types -- classical, rock, etc,) a fine teacher is Marco Accatttis in Jersey City For another approach to guitar, you might try Ed Littman in North Haledon, who specializes in rock and funk guitar. If you're in Warren county, be sure to avoid a place called STUDIO 46. In my opinion this place is a fraud. They advertise themselves as a school but they are not (not accredited, no fulltime staff, etc.). Also the quality of teachers varies quite a bit. Some of the guitar teachers are good, but they have one person purporting to teach violin when he cannot play it himself. (This is the opinion of Anthony Serafini, Adjunct Professor of Violin at Centenary College)

Other excellent piano teachers are Elena Riabinina in Livingston, NJ, Veronica Tapsony in Medord, NJ, Abbey Owens in Upper Saddle River and Youn-Hee Bang in Wayne, NJ Then there is a fine piano teacher, Lan Chakoff also in Livingston, Note that there are also several good online directories of music teachers, such as PrivateLessons.com, and many of the above people can be found there.

Interested in conducting? You might consider Stephen P Brown of Montville, NJ who teaches conducting as well music theory and percussion.
 Conducting Lessons, Percussion

If you choice is a brass instrument there is Richard Bass, who teaches trombone and Gregory London who teaches trumpet in Bremerton, NJ. Also try the Brass Store in Woodbridge, NJ. Also try East Coast trumpet teacher in the Philadelphia/S Jersey area.

     A few tips on CHOOSING a music teacher: tougher than often realized. All too often for example, the teacher cannot play the instrument he/she purports to teach!   There's a very easy way to tell if he/she plays the violin for instance -- look at the FINGERNAILS!  If they look like they haven't been trimmed in weeks, the guy doesn't practice: EVERY serious violinist trims their fingernails every few days. Therefore, ask the new teacher to PLAY something for you. Even a layman should be able to recognized when a performance is inept.
Another piece of useful information is where they received their musical training. If they are a graduate of Julliard, Setnor or the Curtis Institute, those are very, very strong credentials. you are considering learning. A trumpet performance major for instance is a very strong credential. However, other things can easily compensate for a lack of academic background. Long experience in a symphony or band is good.

   For violin repair, best is to send it to Shar Music or Southwest strings; be wary local repairmen. Bob Tice for example takes an inordinately long time and does inferior work.  Also avoid the MUSIC SHOP in Booton NJ; they too take forever to do anything and a crack in my violin was not even close to being repaired properly. Also avoid STUDIO 46 in Hackettstown, at least for string instrument (violin, viola, cello) lessons. They do not have people qualified to teach these instruments. Some of the guitar instructors seem to be good though.

      Also try calling local colleges or universities; even if the instrument professor is too busy they may have exceptionally good students who would make fine teachers.

      Still, even a good player is not necessarily a good teacher: you can next ask for references from other students; a good teacher should not be insulted by such requests.  Does she begin each lesson asking "let's see, what were you working on last week?" A teacher should remember such things AND remember the problems the students were having with the piece. Failure to do so incidates laziness

     Monitor your child's progress. After a few months of lessons playing should be at a reasonably good level If you are not musically knowledgeable, have your child play for someone who is. Sometimes a teacher can play an "advertising" game.... the teacher may have put on a good show for the parents at the first lesson But later, especially if parents are not around, it can happen that the teacher is full of himself, not above insulting both the child's previous teacher as well as the student herself. Needless to say, this is destructive behavior. 

    WARNING TO PARENTS: note that the music world is small, especially within a given state.  Be very, very careful about mouthing off at your child's teacher, arguing over trivialities etc. In an age of email and lightning speed communication  you may find yourself, in effect, blacklisted among all the music teachers in your state.