Winks, 30, joined from Leicester City on a free transfer following the Foxes' relegation, signing a two-year deal alongside fellow new arrival Jacopo Fazzini, who came in from Fiorentina.
He told the club's official channels that the airport welcome left a strong first impression.
"When we arrived at the airport with Fazzini, there were a lot of fans waiting for us. It was great to receive such a warm welcome," he said. "I'm very happy to become part of this club, an excellent Serie A club, with a very good coach."
This is not Winks' first taste of Italian football, having spent time on loan at Sampdoria back in 2022.
"When I played in Italy four years ago I loved the culture, the lifestyle, the football and the tactical side of the game. It's very different from England. Here you use your head more, you think about the game differently," he said. Asked about his own game, he added: "I'm someone who wants the ball on the pitch. My job is to be the link between defence and attack, trying to dominate possession as much as possible during the match."
Winks has previously worked under notable coaches including Mauricio Pochettino at Tottenham and Enzo Maresca at Leicester.
"There are two coaches I've worked with who shaped the way I see football: Pochettino, who gave me the chance to break through at Tottenham — I owe him so much. Working with Enzo Maresca was something that made me see the game differently," he said.
Now he answers to Fabio Pisacane: "Since I arrived here, I've seen that the boss sees football in a very similar way, and for me that's fantastic. I've only been here two weeks but I've already learned so much from him. We're working on how to keep possession, which I think is really important in modern football."
At 30, Winks arrives as one of the more experienced players in a young Cagliari squad, a status he admits feels slightly unfamiliar.
"I don't think I've ever been the experienced one in the teams I've played for before. But it's exciting: there are some really good players here, with a lot of talent, especially among the younger ones. I'll do everything I can to help them," he said, adding that he intends to start learning Italian as soon as possible.
He will also settle into island life with his family in the coming weeks.
"My wife and daughter will come here, they'll live with me in Cagliari. My wife has already told me about the beautiful beaches and how good the food is. I can't wait to learn the language, immerse myself in the culture and experience the city," he said.
Asked to name his best coach, best team-mate and toughest opponent in a quickfire round, Winks answered Enzo Maresca, Moussa Dembele and Lionel Messi respectively.