The Reds, who ended last term in seventh place, begin their campaign at home to Stoke on August 17 and then face a trip to Aston Villa the following week before a clash with long-time rivals and reigning titlists Man United at Anfield seven days later.
Brendan Rodgers' troop will then tackle Swansea (a), Southampton (h) and Sunderland (a) in September and Liverpool legend Carragher - who retired at the end of last term to take up a role as a Sky Sports pundit - believes the matches give the Merseysiders a chance to pick up early points.
"The first five or six fixtures look easier on paper than the ones from last season, and you can see that after the problems with the transfer window last year, they are making early moves [with players]," Carragher said of Liverpool, who have not played in the Champions League since 2010.
"I am sure they will be ready for the start of the season and hopefully they can get three, four, maybe six points from the first two games as they need to get off to a good start and do better in terms of challenging for the top four.
"The last few seasons haven't been good enough but top four has to be a realistic aim for Liverpool Football Club and I think Brendan is raring to go after learning a lot last year."
Luis Suarez's future at Anfield remains in doubt with the Uruguayan continually stating his desire to depart English football and Spanish heavyweights Real Madrid reportedly interested in snapping up the 26-year-old frontman.
Carragher is hopeful that Suarez, who moved to the Reds from Ajax in January 2011 on a five-and-a-half-year deal, will remain a Liverpool player next term, but says that if he heads through the exit door, Rodgers must use the funds to bolster his strikeforce.
"I hope Suarez stays, not just as an ex-Liverpool player but as a Sky Sports pundit," said the former defender. "I want to be watching Suarez and talking about him because he is one of the top players in world football and gives his all on and off the pitch.
"There is talk of Real Madrid but Luis has a long contract and if Liverpool don't want to sell him they can keep him - but if he does go they need to make sure they use the money to get some firepower and have four top strikers."
Manchester United, Manchester City and Chelsea will all head into the 2013-14 campaign with new men at the helm, with David Moyes succeeding Sir Alex Ferguson at Old Trafford, Manuel Pellegrini installed at the Etihad Stadium and Jose Mourinho returning to Stamford Bridge.
Carragher thinks that will make the fight at the top end of the table even more intriguing and is backing Mourinho - who dubbed himself 'The Happy One' after signing on for a second stint in charge of Chelsea - to reignite the Blues' hopes of landing the Premier League title.
"We've known what to expect in the past with Sir Alex Ferguson [at Man United] and Roberto Mancini at Man City for a few years, too, but I think things have opened up now," added Carragher.
"Man United have been very dominant in the past and you'd certainly expect them to be up there again as they have a great manager and a great team, while Man City have been a top side for a few years.
"But Mourinho coming back will add spice; Chelsea haven't really challenged for the title over the last few years but I think that will change."