I know that in the Eastern Orthodox and Eastern Catholic churches communion is given to infants at baptism along with chrismation. What about after that? Do they continue all throughout childhood, or is it just a one time thing until they are older?
In the Orthodox church, communion is seen as a thing to be obtained as often as possible (since it is seen as truly the body and blood of Christ), though not more than once per day. In monasteries there will often be Liturgies every day. In most churches, though, the liturgies will normally take place only once a week. This being said, every Orthodox christian, of any age, will partake of The Holy Mysteries (unless they have not confessed their sins to their spiritual father recently, and think that they are not worthy. Note: At monasteries, the rules for confession may be more strict: confession before every communion, confession within at most the past week, etc.).
So, in the Orthodox Church, most Believers will receive Holy Communion roughly once a week from the time they are baptized, until the day they die. Note: It is seen as a great blessing to receive communion on or right before the day you die.
As far as the Catholic church goes, things are relatively the same; do not receive Communion if you have not confessed any mortal sin you may have done recently, et cetera. They also share the belief in transubstantiation. However, in the Roman Catholic Church, infant and children communion is much less common than in the Eastern Orthodox and Eastern Catholic churches. According to Wikipedia:
Infant Communion (also Paedocommunion) refers to the practice of giving the Eucharist, often in the form of consecrated wine, to infants and children. This practice is standard in the Eastern Orthodox Church and the Eastern Catholic Churches; here, communion is given at the Divine Liturgy to all baptized and chrismated church members regardless of age. Infant communion is less common in most other Christian denominations, including the Latin Rite of the Catholic Church.
For more on the Catholic's view of who can receive communion, see here.