The USA midfielder made the switch earlier this month from Real Betis.
Speaking to the club's media channels, Cardoso beamed: "I'm very excited about everything and very happy. It's a childhood dream that I'm realising and being with the best is always very important to me and, as I said, it's a dream.
"I'm a very laid-back guy. I like being close to my wife and family and enjoying every moment. As a player, I approach every match as if it were my last and try to enjoy it as much as possible.
"It's important and special. I hope to make history for the club, also as the first American here.
"I'm a player who learns a lot by watching others play. Griezmann and Julian Alvarez are phenomenal, we have many players here that I will enjoy every day and will definitely support, growing alongside them."
On coach Diego Simeone, he added: "It's very special, a unique opportunity, and ultimately, in his profession, he is an example. Being able to share and learn from all his teachings every day is truly special."