If I had issues with goiter when I was little, what are the chances my kid will have a goiter?

Depends on cause. Goiter can be due to nutritional/pubertal/environmental reasons or due to genetic reasons. Fortunately, the former reasons can be treated/managed but the genetic ones which are rare cannot be 'fixed'.
Depends on the cause. Goiter can occur in iodine deficient areas but is rare when iodinized salt products are available.Goiter due to immune thyroiditis occurs in family clusters so testing of relatives or kids can spot the problem early, but the problem is not passed like eye color. The relative chances of this for your kid can be better defined if you know what kind of problem led to the goiter.