At what age do boys need to be educated to wear tzitzis?
How careful do the parents need to be in ascertaining that they remain kosher at all times?
The Shulchan Aruch rules (O"C 17:3) that a parent is obligated to educate his child and buy him tzitzit when he is knowledgeable enough to wrap himself in them on his own. The Ram"a adds that he should also be capable of wearing it in a way that keeps two tassels in front and two behind, and to kiss it when he recited Keriat Shema.
Here is the relevant Se'if
קטן היודע להתעטף, אביו צריך ליקח לו ציצית לחנכו. הגה: ודוקא כשיודע לעטוף שני ציציות לפניו ושנים לאחריו (הגהות מיימוני פ"ג), ויודע לאחוז הציצית בידו בשעת ק"ש (מרדכי סוף פרק לולב הגזול).
Today, most children I have seen can dress themselves by age three, and also know how to control their bodily functions, which is why the custom among many today is to begin training a child to wear tzitzit at this age.
I believe the general rule of thumb is that children are to begin observing positive commands as they become mature enough to have some understanding of them.
Regarding tzitzis there is a minhag to have boys wear a tallis katan even from age three, see Sha'arei Teshuvah 17:2, Aruch HaShulchan 17:5 (Children in Halachah 2:iv, Rabbi Simcha Bunim Cohen).