The 23-year-old became Britain's most expensive keeper on Thursday when he completed his £30million move to Goodison Park from Sunderland, and on Friday evening saved a late penalty as the Young Lions drew 0-0 with holders Sweden in Kielce.
"It's good. It's nice to get it (the move) out of the way, and I can concentrate on the Euros and go as far as I can with England," he said on Sky Sports. "Like I say, I can relax now, not worrying about anything.
"It's a great club to go to for myself, to move that next step on my journey. I feel like Everton was the best place for me so it's a good one for me."
Jordan Pickford of England saves a penalty shot from Linus Wahlqvist of Sweden during the UEFA European Under-21 Championship match between Sweden and England at Arena Kielce on June 16, 2017 in Kielce, Poland.
Image credit: Getty Images
Group A also includes hosts Poland and Slovakia and, with only the three pool winners and best runners-up going through, England must improve.
But Pickford was content enough with the outcome in Kielce.
"The main thing was not to lose the game," he said. "It would've been nice to get the three points. Winning the first game's always nice in tournament football. But we've come away with the draw, and it's down to us to win the next two."
On his spot-kick heroics from Linus Wahlqvist's effort 10 minutes from time, he modestly added: "I just waited a bit longer, held my ground, dived to the left and got my top hand to it. It's what you do."