Birth records are valuable document in our family searches. The primary location to locate these records is at the county courthouse in Painsville. It did not become the responsibility of the county to start recording birth records until 1867. Prior to this time they are non existent unless found in other sources.
Birth records start in 1867 and run until present. Be sure to have at least a decade narrowed down when you begin your search. Birth records tend to be indexed so you will be able to not only get the name of the person you are looking for, but also people with the same last name. This is helpful is rural counties where the population was not so high. Sure there are exceptions to this search, but they are few. Common surnames in a particular area translate into common relationships.
Be patient with the courthouse folks and realize they get many requests for searches. If you plan to go in person it is always a good idea to call ahead and make sure that they will be open when you wish to go.