Pupils as young as five should be given lessons in how to use the internet safely, urge campaigners.
The call comes amid fears that children are increasingly exposed to online pornography and forced to send indecent images of themselves to others.
The charity ChildLine is holding assemblies in every UK primary school to tell children how to stay safe in the digital age.
Claire Lilley, of the NSPCC, said: "We are facing an e-safety time bomb."
Ms Lilley said online abuse was one of the biggest child protection issues of our time.
"Young people tell us they are experiencing all sorts of new forms of abuse on scale never before seen.
"The internet and mobile phones are now part and parcel of young people's everyday lives. They are the first generation who have never known a world without them.
"The benefits are huge, both socially and educationally, but so too are the dangers."
ChildLine says its helpline took some 3,745 calls about abuse via the internet and mobile phones last year, with most callers aged between 12 and 15.
Some 250 callers said they were being groomed for sex online.
There was also a 70% increase in calls about online pornography with 641 calls, some from children as young as 11 years old.
The campaigners also highlighted previous research which has shown that many teenagers see "sexting" and hard core pornography as "mundane", while cyberbullying is a growing and insidious problem.
On Monday the Child Exploitation and Online Protection group (CEOP) warned that paedophiles are increasingly forcing children to carry out sexual acts online.
The NSPCC says that schools need to step in as the issue is something that parents struggle to keep up with.
It is calling for lessons in all schools on internet safety with young people themselves sharing tips for keeping safe online and more advice for parents on how to talk to their children about the issue - just as they would about being wary of drugs or strangers.
It also wants all internet service providers to provide easy systems to allow parents to install online blocks and filters in their homes.
However some experts argued against blanket filtering of the internet. Phil Bradley, of the Chartered Institute of Library and Information Professionals, described it as "like using a hammer to crack a nut".
He warned that blanket filtering could have unintended consequences such as blocking legitimate websites on sexual health and identity.
"When it comes to the internet... children need to learn how to use it safely and how to differentiate the good from the bad."
A government spokeswoman said web safety was taught in schools as part of the personal, social and health curriculum but it shouldn't just be left to teachers.
"Just as parents would teach their children to cross the road safely, they should also help them learn how to stay safe online.
"The industry also has a vital role to play. Websites children use to network should be fully moderated and internet companies should provide parents with all the tools and information they need - including access to parental controls."
Tips for promoting internet safety are available online on Tuesday, 5 February, which is Safer Internet Day