The 25-year-old made his competitive debut for United in Saturday's 1-0 win over Tottenham to open the league season following a £25 million summer arrival from Southampton.
The move marked quite a rise for Schneiderlin, who played in League One, the Championship and top flight for Southampton and is now expected to make his Champions League debut on August 18.
"All my life, I have been working for this moment to happen," he told reporters. "I could have gone before, but I was not physically ready when I was younger. But now, at 25 and with all the experience I have behind me, it's a good step for me.
"Of course, it's different here and you play with more fans, but after that, it’s a football game. Of course there is more expectation on you, more focus on your performance, but that doesn’t scare me.
"I know that I'm playing with good players, that I'm at a good club with a good structure. I can't wait to show what I can do."
Schneiderlin said playing outside the top flight helped him improve as a player, having suffered relegation to the third tier before two promotions with Southampton saw him reach the Premier League.
"Southampton toughened me up, physically and mentally," Schneiderlin said. "When I came to England the target was to play in the Premier League and to get back into the Premier League the next year when I was 18.
"Going into League One could have made me sad or worried about my future. But I was strong mentally and I was surrounded by good people who made me understand how I have to get better, which part of my physicality I have to improve.
"I'm really grateful for that. I worked really hard on my fitness and the way I understand the game. Playing in League One against strong physical players can only help."
