"We can go as far as we want to," Pickford, who joined Ronald Koeman's side from Sunderland, told the club website (www.evertonfc.com).
"We've got the team and we've got the depth as well, which is massive, especially in the Europa League, but also week in, week out in the Premier League."
"The club is showing massive ambition and that's one of the reasons I came to the club."
Pickford said he was excited to work with England's record scorer Rooney, who returned to Goodison Park after 13 years at Manchester United.
"You've got to look up to him (Rooney), like you look up to so many other people when you are wanting to become a footballer," Pickford added.
"But it's not just Wayne. The manager has brought in a lot of good players this summer, and there were a lot of good players already at the club, so it's an exciting time to be here."
Everton will begin their new season against Ruzomberok of Slovakia or SK Brann of Norway in the Europa League third qualifying round. Koeman's side will be at home in the first leg on July 27 with the return fixture scheduled on Aug. 3.
(Reporting by Hardik Vyas in Bengaluru)
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