Is there a program or a software for a small business accounting for a non accountant person?

There are many books.
There is software such as Quickbooks (several variations for different kinds of businesses and accounting).
I would get both, especially if you are brand new to having a business. I would go to a library or bookstore so that you can leaf through them. Many, many differences.
It helps to read irs.gov as you go too, because of what you need to do to keep records in a way that makes filing taxes easy.
(Your first year as a small business, you do not have to file quarterly estimated taxes.)

You could try Quickbooks as has been suggested or MYOB