Well, I have to laugh because of your comment on ballet. So many people think that being in a setting with 20 kids in the room (such as PUBLIC SCHOOL) equals socializing. Atleast you realize what socializing is, and that it does not happen in that room. It does, however, give the child an opportunity to make friends and socialize outside of that setting-such as inviting the child for a play date, birthday party, etc.
Anyway, my daughter is ten and my son is seven. They have always been homeschooled. They also have a six month old baby sister.
First, they are of course around their cousins and the children of our friends regularly. This is one of the more social options.
While we do not have neighbors like most people do (we live in the country), most children can go outside and play with the other children in the neighborhood as well. Our children love to go visit one set of neighbors, an older couple with grown children and grandchildren. The neighbors on the other side is a family: preacher, wife who is a nurse, and their three children (they are all older teenagers). Another neighbor is a friend of my husband's, and his son just recently moved back home after a divorce (he is a family friend as well). The older couple recently built a cottage on their property to rent out for retirement income, and we have not met the new occupants of that yet. Down the road is another couple who own cows and horses, we walk down there often to observe them. The man was my husband's shop teacher in school. My children speak to all of these neighbors, even though they are not children-and that is socializing.
Church. This is another highly social opportunity. Our church has traditional services on Sundays and then a children's program on Wednesday evenings. We often stay late while the children play with other children.
Scouts. Both children are in scouts, and this is highly social. They have time for free play, they do activities, etc.
Baseball. Both children are on teams in our community. I would not have thought that this would have been the social opportunity it is. They (and my husband and I) have made new friends and met a lot of new acquaintances. My son met his best friend-a private school student-at his first tee ball practice. They just hit it off from "Hi, my name is...".
They have been in, are in, or will soon be in: Golf, Swimming, Tennis, Bowling League, Chess Club, Drama, Gymnastics, Karate. These are some of the activities that were more "meet for the activity, and then it is over" for us, but they did make acquaintances and friends and had social experiences (such as dealing with bad behavior).
They are in 4H, another very social opportunity.
We visit the playground often, and if we are one in our small community they usually run into someone they know. Whether we are at one near our home or one elsewhere, they always make new acquaintances.
They attend camps, clubs, in our community.
We are in a support group with other homeschoolers. They get together with others regularly for field trips, play dates, science fairs, spelling bees, etc. VERY SOCIAL.
They also have the opportunity to socialize in the real world. They speak to the librarian, the butcher, the doctor, the lady with the new baby in the line behind us at the store, the elderly man with us in the elevator, the mail man...
There is no lack of socialization amongst homeschoolers. The term "home" school does not mean that they are at home all of the time. We are busy out in the real world instead of just reading about it in books. The home just serves as our base, much like the "homeroom" in public schools.
I hope this helps you, and I wish you must joy (and patience) on your homeschool journey.