Liverpool lost out in last year's League Cup and Europa League finals and Milner, who won the FA Cup with Manchester City in 2011 before their run of two league titles in three years, said winning one trophy is a catalyst for more.
"Personally the biggest thing for us at City was winning the FA Cup: getting that first trophy under our belt, winning something, giving you that belief," Milner said.
"Champions League was a step along the way. Last year we came close (to a trophy) but couldn't get over the line. But playing in those sort of occasions and in those big games is important.
"Hopefully now we can stick a few trophies in the cabinet. That is the next step."
Midfielder Adam Lallana also expected Liverpool to be ready to compete for silverware next season.
"It has been a good season, not just myself but I think the team has come on leaps and bounds," Lallana told the club's website. (www.liverpoolfc.com)
"There is still a lot of room for improvement and hopefully we can continue that in the Champions League next season.
"We've got top four now... Hopefully we can add to this great squad and really compete for silverware next year."
(Reporting by Aditi Prakash in Bengaluru; Editing by Peter Rutherford)
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