The new rules make it illegal for children to buy tobacco and nicotine vapour products (NVPs).
However, campaign group Action on Smoking and Health (ASH) Scotland gave the new laws only a cautious welcome.
Chief executive Sheila Duffy said the priority should be to target retailers who sell traditional cigarettes to children.
Under the new laws, anyone buying NVPs for those underage will be breaking the law.
And shops selling the products will be required to have an age verification policy and to be registered.
Public health minister Aileen Campbell said: "We know e-cigarettes are almost certainly safer than cigarettes and have a role to help people quit smoking, but we don't believe children should have access to them - that's why these age restrictions are so important."
She added: "We are working closely with the Scottish Grocers' Federation to make retailers aware of these changes and what they mean for their daily business.
"A campaign is already under way across Scotland and will continue to run throughout the summer to ensure everyone is aware of these changes to the law."
Ms Duffy, of ASH Scotland, said: "Nicotine is addictive and as there is some level of health risk associated with using these products, it is right that we keep them out of the hands of children.
"With the regulatory framework for e-cigarettes falling into place, now is the time to remind people that smoking tobacco is by far the more harmful activity."
She added: "There are still 30 to 40 young people in Scotland taking up smoking every day and the priority must be for further action against those retailers who sell to children and to challenge the attitudes amongst adults who buy cigarettes on heir behalf."
The changes were brought in by the Health Act 2016, which also set out restrictions on e-cigarette advertising and a ban on vending machines selling the products.
Both measures are due to be introduced later this year.
Restrictions on e-cigarettes, including a ban on their sale to under-18s, have come into effect.