The Wales international completed a £15 million move to Liverpool on Friday to secure a reunion with his former Swansea City manager Brendan Rodgers.
Allen had played a part in the Swans' promotion to the top flight and was a key figure in the club's impressive debut campaign in the Premier League.
The 22-year-old is looking forward to a new challenge and acknowledged his gratitude for the success he has enjoyed to date in his fledgling career.
"The last couple of seasons have definitely been a whirlwind," Allen told Sky Sports News.
"First and foremost, we got promotion to the Premier League which was something I hadn't imagined so quickly and then off the back of last season to find myself where I am today has been an incredible journey.
"I'm very grateful for the times I've had and I'm looking forward to more magic moments in the future."
Allen is aware of the competition for a spot in Liverpool's starting line-up and he is aiming to swiftly slot into Rodgers' plans at Anfield.
He added: "There is competition for every single place in the team, it's full of world-class quality.
"I'm really excited to be playing with the likes of the players who are at this football club and hopefully I can gel straight with them."