Prince William said the island had been a "special place" during his work as a pilot at RAF Valley.
"I thought search and rescue duties... were physically and mentally demanding but looking after a three-week-old baby is right up there," he said.
He had been giving a speech at the island's annual agricultural show.
It was one of Prince William's first public engagements since the birth of his son on 22 July.
He has been based at Anglesey since 2009 but is due to leave next month.
In a speech, which he began giving in the Welsh language, he thanked the people of Anglesey for being so welcoming to him and his wife.
Speaking about the Duchess of Cambridge and his son, Prince William said: "She and George would have loved to have been here. He's pretty loud and of course extremely good looking."
He told the crowd: "I thought search and rescue duties over Snowdonia were physically and mentally demanding but looking after a three-week old baby is right up there.
"I know that I speak for Catherine when I say that I have never in my life known somewhere as beautiful and as welcoming as Anglesey.
"I know that both of us will miss it terribly when my search and rescue tour of duty comes to an end next month and we have to move elsewhere.
"From the bottom of my heart, thank you for making my wife and me so welcome when we arrived here, as you do thousands of visitors each year.
"This island has been our first home together, and will always be an immensely special place for us both.
"Catherine and I look forward to returning again and again over the coming years with our family."
The duke drew roars from the show crowd when he began his speech from the main ring in Welsh.
In translation, he said: "Thank you to the people of Anglesey. It's pleasure to be here. I'm so glad to have lived on Anglesey - the mother of Wales."
Continuing in English, he joked: "My Welsh pronunciation isn't great yet, but it is certainly getting better I hope."
Earlier, he was asked by one show visitor about his son, the duke replied that Prince George was fine but "has his moments".
The prince was introduced to civic dignitaries, including the island's newly elected-assembly member Rhun ap Iorwerth, before being shown a display of gundogs.
The prince was then presented with his very own falconry glove during a display by Sophie Large, 11, from Wrexham, who is the daughter of falcon master Terry Large.
She then helped the prince take part in a display of Harris Hawks.
"It was very special, I know it is privilege to be able to meet him," she said.
"He was kind and very nice."
Caroline and John Roberts from Llanddaniel Fab both spoke to the prince.
"He was lovely, a real gentleman," said Mrs Roberts.
"We asked him how the baby is - he said he was fine but had his moments."
Well over a thousand people gathered around the cattle show ring in a bid to catch a glimpse of the prince, where he chatted to judges and those showing off their animals.
He then took a break from the throng to meet members of the island's Young Farmers Club, which has its county office on the showground.
The visit is expected to mark the duke's farewell to Anglesey at the end of his tour of duty at RAF Valley.
The prince has kept the crowds at the show happy - devoting plenty of time to stop, chat and shake the hands of the hundreds of people gathered around him.
It took him over half an hour to make the short walk from the country sports field to the main part of the showground, stopping on his way to admire vintage tractors, and to chat with a motorbike display team.
Louise Roberts was on holiday with friends from Liverpool, and had come to the show unaware the prince would be there.
"He shook all our hands, and the children, it's been brilliant," she said.
"He's really taken his time to stop and speak to people - it's wonderful."
With the prince in a green country jacket and casual trousers, the royal visitor has had a very warm reception.
"It's all very calm and relaxed, everyone is in a good mood, and there's no pushing or shoving to get to see the prince," said Janet Doyle, who was with Mrs Roberts with her family.
"That's probably why he's spent so much time with everyone.  And my children have had the chance to shake the hand of someone who will become their king one day - that's something special."
The showground had swelled with crowds ahead of the royal arrival, with hundreds already milling around by 09:00 BST.
The two-day event, which ends on Wednesday, has featured a collection of livestock of up to 300 cattle and 800 sheep.
It has also hosted 350 trade stands displaying agricultural products, clothing, crafts and vehicles.
The visit is likely to be one of the last public appearances on Anglesey by the prince before his three year posting at RAF Valley finishes at the end of September.
It is widely thought he will return to London for a posting with his regiment, the Household Cavalry's Blues and Royals, which will allow him to carry out more royal engagements in support of The Queen.
The Duke of Cambridge has spoken of the challenge of being a father of a three-week old baby during a farewell visit to Anglesey.