The Duchess of Cambridge, as she is officially known, started the "5 big questions on the under 5s" initiative in Birmingham in central England and will visit Cardiff, London and the county of Surrey, south of the British capital, on Wednesday.
The tour comes as William's younger brother Harry flew back to Canada after sparking a crisis in the British monarchy by announcing he and his American wife Meghan wanted to reduce their royal duties and become financially independent.
The issue has dominated headlines in Britain over the last fortnight and culminated in a deal which means the couple will step down from their royal roles to spend most of their time in North America.
Meanwhile, William, 37, and Kate, 38, have continued with their official engagements, in keeping with the royals' determination to carry on with business as usual.
On Tuesday, William attended a meeting of the United for Wildlife Taskforces, which aims to tackle the illegal trade in animal products, while Kate was at Birmingham's Science Museum to launch the survey designed for people across the country to give their views on children's early lives.
She will follow up on Wednesday by visiting a nursery for breakfast, attending a baby sensory class and then traveling to a women's prison to talk to inmates.
The aim of the survey, which contains five short questions such as "What do you believe is most important for children growing up in the UK today to have a happy adult life," will provide a vital source of information and help bring about positive, lasting change, her Kensington Palace office said."This survey will give the early-years sector valuable direction in designing and delivering services and support which reflect what matters most to people," said David Holmes, chief executive of the Family Action charity.