Following the win over the Austrian outfit, he told reporters: "Every player is impressing me. From when I played against Real Madrid, [I knew] all the players are world class. But just the way the lads are; when you come in, you expect it to be very intimidating, you don't know what to expect.
"But it was very warm, very welcoming, they made me feel at home straight away, which was amazing. It allows me to play as well as I can and today coming into the game, I was confident to go and perform.
"The settlement period now is done. It's about kicking on and being the best player that I can be and giving the best for the team and helping the team win as many games as possible starting with this competition."
The defender lifted two Premier League titles, the Champions League, the FA Cup, the Club World Cup (in its previous format) and two League Cups at Liverpool. He also notched a remarkable 67 assists during his time at Anfield, a Premier League record for a defender.
Alexander-Arnold made a positive impression on his team-mates and fans by speaking in Spanish at his unveiling. However, he concedes that mastering the language remains a challenge.
He added: "Of course speaking and being a part of the conversation is harder. You work hard to try and understand what people are saying and be a part of it.
"But the manager speaks to me a lot individually to make sure I understand everything that gets said. The meetings are in Spanish so he goes out his way and the staff go out the way to make sure I understand the game plan. So there's differences but it's just football."
Real progressed to the competition's last 16 as group winners. They began the campaign with a 1-1 draw against Al Hilal of Saudi Arabia, before defeating Mexico's Pachuca and then Salzburg. They will next face Juventus, who suffered a 5-2 defeat at the hands of Manchester City in their final group match, on 1 July.