Ditto, on all counts.
I will offer this additionally: be cognizant of one dynamic when finding/observing a school/instructor: "empty your cup", so to speak. By that, I mean try not to have a preconceived notion of what you want and, hence, judge based on that. Watch the dynamic of class, how the instructor teaches and interacts with the kids, how the kids seem to react to the instructor, etc.
I've had parents not sign their kids up with me and then tell me, when I follow up, that they thought we were too lax with discipline. They were apparently looking for a facility run like marine boot camp, where if a kid leaves line they get yelled at and are made do 50 pushups. Then I've also had parents tell me they were not sure their child could handle the structure and discipline. I can tell you who runs those households. I take all that to mean we've struck a happy medium!
A few thoughts (I've just completed two two-day special camps we do in July for our youth students; a couple dozen kids or so in one gym, for 8 hour each day). Whew! So, I'm weary of body and mind, but fresh on comments.
Here are some off the top comments on youth classes: (with my unique bullet points)

Does the instructor(s) treat the kids respectfully?

Does the instructor(s) seem to enjoy teaching kids? Is he/she having fun? (I do!) (Granted, new instructors can find teaching kids to be tedious, or have some trouble managing kids, so forgive them that a bit. It is challenging!)

Does the instructor encourage?

Does he/she ensure all kids get attention?

Does he/she push the kids to behave, to learn, to pay attention, to get better? It is, after all, martial arts and not day care play time. Kids can have fun, be treated like kids (not marines, but not hapless babies, either), but be pushed to toe the line and really learn stuff. (And, yes, I will occassionally, VERY sternly address a misbehaving student when necessary.

Is he/she not overly lax or overly stern (watch a few instructors; you can figure this out).

Does he or she (or the school) seem to operate with a plan or curriculum? Does he/she seem relatively ready, prepared, organized?
Regarding the students, most of what you want to see is implied in the above, but I'll stick a few down here:

Are they having fun? There should be some smiles in there.

Is anyone getting neglected? Are one or two stars getting most of the attention? (Some students do need extra attention at times, but I turn it into a "today we really have to help Miss Denise learn her new kicks, so we're going to all work on that together and all get better at it . . .)

Are they really learning some technique, or just playing games? Even a 50-50 split is OK, and some games/activities are teaching some skills/principles. Still, there should be an obvious balance. They don't need constant military death techniques, but they should have real martial arts learning in there, otherwise what are you paying for? IMO, a student, even a very young one, should be able to both learn new skills/technique and get better at performing them each class, even if in a small way.

Are they learining good messages of confidence, manners, self-control, self-discipline, citizenship, etc? I talk about these types of things to students throughout class while training. Everything they do is a lesson!
My bias: because of the natures of martial arts in the US, the prevalence of Taekwondo and youth instruction, credentials, certifications, etc, if there is a Taekwondo school in you area, that is a good place to start. Lots of americanized karate schools are more a franchise and a business than a solid martial arts school, IMO (and the opinion of many other). But, there are good ones. Like Kuri said above, it ultimately comes down to the school.
Further, if the Taekwondo school is WTF (World Taekwondo Federation) affiliated, that's potentially another feather. Important: In a Taekwondo school, ask if their instructors are Kukkiwon (the world taekwondo headquarters in Soeul) certified. They should be.
In any school, ask about training, credentials and certifications, lineage of instructors/authority/oversigh t, etc. Try to get a good sense of if what they tell you means anything of substance.
Whew. That's all, just to get the ball rolling. Feel free to post back or email/PM if you'd like. Good luck!