My husband grew up in a family of musicians (professional and not) so we had some specific things we wanted our kids to get out of music lessons - mostly just loving it and not having it be a drag to practice, but also a strong foundation in music in general so they could branch out into whatever area they wanted later. DD (almost 16) plays piano, guitar, ukulele, a little flute and sax. DS (almost 13) plays piano, bass and some drums. They both sing parts too, and now that DS is into it, we can sing 4-part Christmas carols! Here are some things we've learned over the years (I'm not a musician, by the way, but I started taking piano lessons a few years ago just to be able to participate with my kids - very difficult as an adult! But it's nice when they help me.):
Lots of musicians say piano is a great instrument to start on because (a) it's very visual - you can see the relationships between the sharps, flats, octaves, and key signatures, etc. right in front of you, (b) you press a key and get the note, unlike the violin, for example, where you need a lot more practice to get a nice tone and the right notes, and (c) you learn the bass and treble clefs at the same time (also includes chord structure and harmony), whereas a violin or flute only uses treble.
A plus for piano, guitar and ukulele is you can accompany yourself singing on those, but pianos aren't portable.
I think the teacher and teaching method is critical for very young kids starting out - it has to be fun, low pressure, and leveled to the child's attention span and interests. We started the kids at age 5 with 30 minute one-on-one piano lessons with an 82 year old lady who taught them "songs" (really just a modified scale with words) right from the first lesson. This was perfect when the kids were younger, and later when they transitioned to a more traditional approach, we found they really had absorbed a lot of theory - they just had to learn the terminology!
People I've spoken to who have used the Suzuki or comparable teaching methods say that it does work, however they learn How to Play an Instrument, but not necessarily how to relax, mess around, get loose. And the structure doesn't work for every child or family.
We are trying to balance really learning technically how to play/music theory with relaxing, participating with other people, enjoying the experience, and not having it have to be perfect. It doesn't always work out that way (like in the couple of weeks before a piano recital) but generally it seems they enjoy being able to play!