I think this means, I will never again [nie wieder] go to childcare.
In English, likewise in many cases in German, the proposition to be used with to go (gehen) is to (zu).
So why is in die Kinderbetreuung instead of zur Kinderbetreuung?
zum is short for zu dem. This would assume that Betreuung is masculine, but it is not. So zur (short for zu der) would be a correct choice:
"zur" is just as good as "in die". I've experienced that in this context "in" has become the preferred preposition like in "in die Pflege" geben. That might have to do with the fact that somebody is being handed over into the responsibility and care of somebody else, in German in die Obhut geben. So maybe people might be distinguishing this from a casual visit to a, say, a friend.
Both versions are correct, depending on the geographic region where you are:
If you are in Germany, you say
Because when you say this in Germany, you are meaning "I go to the institution and I enter the building where it is located". When you say this in Austria, Austrian people will understand "I go to the building and I stop in front of the door. I do not enter the building."
I am born in Austria and I live there. When I was a young child (5 or 6 years old) I never did understand, why german kids say that they go to their school, but why they never was talking about going into the school. I really did believe (but not for very long time), that german schools only practices outdoor-education.
So, if you are in Austria, you explicitely have to say that you go into that building. So in Austria you say:
The difference comes from different ways of thinking about institutions and their buildings: In Germany a childcare or school is a special kind of institution (necessarily located in a building). In Austria we think different: A school or childcare is a building (hosting a special kind of institution).