Qualifications
Are you patient? Kind? Very patient? Good with children? Flexible, but not a pushover?
Excellent. Basically, to be a great daycare worker, you just have to "get" kids. A lot of the things they do to test you are simply because they don't know any better. Or, you give them instructions, but they don't understand so they do something else. Usually, they aren't doing it just to make your eye twitch.
Kids are emotional and you have to know how to handle their pint-sized drama, because to them, it's huge. Remember, you're teaching them conflict resolution and how to see another's point of view. So a quick, "OMG...just stop," probably won't fix the problem. Children require lots of modeling.
In other words, you need to show them how to behave, not just tell them. Even if their drama is cute to you, to them it's real. Respect that and gently guide them to a more mature perspective. Or bribe them with treats. (Not really. Well, maybe.)
Whether you do any post-high school education or not, you need to pass the background check and get CPR certification. Also, keep in mind that once in a while you'll have to talk to parents about something awkward involving their child. Maybe they won't stop hitting another kid or they're teaching the whole class to swear. So on top of that background check, you'll need good people skills, good communication skills, and the ability to work with a very diverse group of people.