The Merseyside outfit finished a disappointing eighth last season, despite substantial investment in the likes of Stewart Downing, Jordan Henderson and Jose Enrique.
Adam also joined the Anfield outfit last summer from Blackpool, but saw his season come to a premature end through injury.
A League Cup win was not enough to save manager Kenny Dalglish from the sack, as Liverpool ended the campaign 17 points adrift of fourth-placed Tottenham Hotspur.
However, the Scotland international believes the Reds have the quality to secure a spot in the Champions League next year.
"We should be challenging for the Champions League, no doubt about that," he told the club's official website.
"You set your standards high and we've got a good group of players. People saw that last season and at times we were unlucky.
"We need to set our standards as high as we can, participate in every tournament we are in and try to win everything available to us."
A knee injury cut the 26-year-old's debut campaign with Liverpool short, but Adam is determined to prove his worth to new manager Brendan Rodgers, starting with their pre-season clash with Toronto FC this weekend.
"I'm fine now, I was back training in the last week of the season basically and I had four weeks' break in the summer. I'm back now and I've joined in every session since the lads have been back," he added.
"It's good to get going again because the last couple of months were frustrating but it's time to reflect on what you have done and it's good to be back in the groove again.
"It has been great for us to come here and get the work done in good conditions.
"The first 10 days of pre-season at home was a bit different, with the rain and the wind, but us coming here is good - and good to get some warm weather and put the good work in.
"For me it's about getting fit again and trying to get some minutes back on the pitch.
"Missing a couple of months at the end of the season was difficult so getting back on the pitch is probably the biggest thing for me, and hopefully showing the new manager what I can do."
