let's talk about what all students will need, and then we'll get into more specifics for younger students.
Music is a language, and the best way to learn language is to practice it daily over a long period of time. You can't practice piano without daily access to one!
If, because of budget or space limitations, you need a keyboard instead of a piano - that will work at the beginning levels. When I started piano at the age of 14, I had a 61-key keyboard for the first year and a half. We only got a piano after it became clear that I was very interested in continuing the study of it.
All people need to practice skills to get better at them. Some people start out with earlier success in piano - but I'll take a diligent practicer over a lazy prodigy any day.
If you're brand new to piano and you want to be able to play a variety of pop tunes, with daily practice (30 minutes or so) you can probably get there in a couple of years.
(This is different than learning a single tune by rote. Of course you can go find a YouTube video that shows you exactly what notes to play and in what order, in a few weeks - but do you really understand the music? No. And you can't carry those skills to any other song, because you don't understand what you're playing.)
If "a couple of years" sounds like a long time to you, just remember - that time will pass whether or not you start today. What if you had started two years ago?
This will help your instructor to build a plan specifically for you. Although the basics are all the same, I'm going to introduce classical music to a student interested in classical, and pop / blues / jazz to students interested in those genres.
(Young kids usually don't have a sense of what kind of music they like - or else it's the music the parents listen to. Personalized tastes in music usually don't develop until the teen years.)
Mostly, someone who is patient and kind, and who has good chemistry with the student.
If they have a degree, that's great. If they don't, their professional experience can be every bit as valuable.
There are people who have Ph.D.s in music but can't relate to a five year old trying to learn their first notes, and there are people who can play a piano like you've never heard before, but can't teach to save their life.
Most piano teachers should be willing to offer a trial lesson, either as a paid one-off lesson or as a freebie. You should feel free to try out a few different teachers before committing to one. No teacher will be the best fit for all students.
If you're looking for piano lessons for your kids, there are a few things to consider.
Students see their piano teacher once per week, usually. Then they're sent home and told to practice their lesson every day (and hopefully the teacher has done a good job of showing you exactly how to practice.)
For very young children, it'll be your responsibility to help them with their practice, just as you help them with their homework. This means you have to understand the assignment well enough to help them (which should be discussed at every lesson).
Eventually we want them to be able to practice on their own, but at the age of 5 or 6, kids think sitting down at the piano and playing a few notes for five minutes constitutes practicing. It does not. (Usually 15 minutes a day is a good starting point for young beginners.)
Sometimes they don't know. Fair enough - sometimes it's a good idea to commit to at least one semester and decide then whether they want to continue or not.
Sometimes you as the parent want them to learn piano just for its own sake, or the benefits it offers in mathematics and spatial reasoning, or to appreciate music on a deeper level. Fair enough - but understand that some kids are strongly opposed to it and will have emotional meltdowns during lessons and practice, which may or may not be overcome through time.
Sometimes your child may express strong interest in a musical instrument other than piano. My personal recommendation in those cases is to go with whatever instrument they're really interested in - but keep them committed a semester at a time.
I and most of the piano teachers I have met will privately teach students as young as 5.
Students at age 3 and 4 usually will benefit from a group musical setting more than private lessons.
Even at older ages, some students find it extraordinarily difficult to sit and focus for more than a few minutes at a time. As lessons are typically 30 minutes, and practice sessions should normally be 15-30 minutes daily, if your child has problems focusing for this long, you may want to look into piano teachers who specialize in shorter attention spans.