Los Rojiblancos sealed a €25m deal for the France international from Real Sociedad earlier this summer after the 23-year-old helped his nation reach the quarter-finals at the World Cup in Brazil.
Atletico coach Diego Simeone was full of praise for Griezmann, along with the club's other new signings, after they beat fierce rivals Real Madrid in the Supercopa de Espana on Friday.
And following the victory at the Vicente Calderon, Griezmann admitted that he was starting to slowly find his feet in the Spanish capital.
"I'm understanding the game. Little by little, I'm going to gain confidence," Griezmann said.
"Hopefully I can give more. I'm very happy, happy to be at this club."
Griezmann also acknowledged the "incredible" nature of the Atletico fans, which he witnessed during the 1-0 win over Madrid - a result secured by Mario Mandzukic.
"Tonight they [the fans] have shown [how incredible they are] again," he said.
Atletico kick off their title defence when they play their first game of the new Primera Liga season away to Rayo Vallecano on Monday.
Despite having lost several key components – including Diego Costa and Filipe Luis – from their title-winning team of last season, Griezmann insists that Atletico are focused on playing well and racking up the wins in a bid to defend their La Liga crown this term.
"We'll see, we have just played two games, now we are quiet with our feet on the ground and moving forwards."