After how many days is it safe for my child to attend school without being contagious to others? (older school going child)

Simple answer: 24hrs. The common advice and the policy in most schools is 24 hrs without fever. However, the truth is that with some viruses, one can continue to shed virus particles for longer. This includes coxsackie virus, which we see in epidemic proportions every summer because of prolonged contagion.
Variable. The time varies depending on the disease. A good general rule is after the child is fever free for 24 hours and without symptoms or on an antibiotic for at least 24 hours and fever free.