As a child I was brought up a strict catholic and I had to attend catechism classes every Saturday morning at our church. I remember taking first holy communion when I was around 7/8. There were four of us from the class taking it at the same time and all we had was the church service with family members and a small celebratory gathering with a buffet in the church hall afterwards.You can have your First Communion at any age. Either you're a cradle Catholic and you receive it around the age of 7 (i.e. when your parents think you're ready) or you come into the Church and receive with the other communicants after your baptism.
I've never actually heard of a full blown First Communion party, though! The most we had was going home, opening presents, and eating cake.