So it turns that in our state I need supervision. I could go with umbrealla type religous educ. type of place or my local school (?). What are the implications of each choice? How would you explain them in more details.What is better and why?
Homeschooling and supervising.
I think each of these programs is different everywhere. Where I live, the public school thing in my district for homeschoolers works like this: Child is enrolled in the public school, child attends one day of classes each week--mostly enrichment type, parent chooses curriculum, school has small budget for non-religious curriculum, parent submits monthly review online, parent and child meet twice a year with assigned teacher (with some samples of work), the child takes the annual assessment that the state requires. The school also has extracurricular extras (optional): fieldtrips, music classes, gymnastics, etc. available. Currculum that is not consumable goes back to the school when you are finished with it; the school maintains a curriculum/resources library for parents to use.
Our state also has a couple virtual schools. For these, your child is still a public school student. Your child does all learning at home. Each one is done differently so it may be best for you to see if there is a virtual school in your state. I am in WA state and we have the choice of K12 WAVA, or CVA (columbia virtual academy). With CVA, the parent can still select child's materials. I don't know about WAVA. With CVA, child has weekly contact via email with their assigned teacher, parents submits monthly reports. CVA will purchase materials and pay for certain lessons for child. CVA has fieldtrips available. If you are enrolled full time, you are a public school student and must take the annual state assessment. If you have an IEP it comes with your child.
What state are you in? Perhaps someone can tell you about their experience within your state.