The Magpies' 65-point haul in the Premier League saw them qualify for the Europa League for the first time in five years.
Coloccini, 30, is now on a mission to bring Champions League football back to Tyneside.
"I accept the Premier League is becoming far more competitive and challenging at the top end of the table is becoming increasingly difficult," he told The Sun. "But we are confident that more success lies ahead. Newcastle are one of the biggest clubs in England and therefore we should be challenging for major honours.
"We don't feel under any more pressure because of what we achieved last season. If anything, we are relishing the season ahead. Our success has brought with it greater confidence and more determination to do better.
"Last season we more than demonstrated we are a side that can challenge the bigger teams. Up to the last day of the season we were challenging for Champions League football. It wasn't achieved by chance, either. Over the season we showed great consistency and our goal this season is to better that success."
Newcastle will begin their Europa League campaign in the playoff round.