Sturridge scored 21 Premier League goals last season, 10 fewer than the now-departed Suarez, who has signed for Barcelona.
The 24-year-old said it was time for him to improve further and others to deliver, with the likes of Lazar Markovic, Rickie Lambert and Adam Lallana added to the attack in the close-season.
"In terms of next season for the goals, I'll step up myself and everyone has to," Sturridge told British media.
"It was a unique partnership and I wish Luis all the best. We shared some good times together.
"We created history with our goals and became the most successful Liverpool partnership in one season.
"But we will move forward as a team. It's everyone's job to do that, the new boys who've come in as well as the players who have been there all season."
Sturridge, whose team were league runners-up behind Manchester City last season, said the additions by manager Brendan Rodgers had him excited.
"Liverpool is a huge club, an elite club and people don't realise that," he said.
"This is a huge club, with a lot of players who can step up. Luis has left. That's football, he's moved on.
"For me, with the signings the manager has made - exciting players, young players, up-and-comers as they call them - we are going to have that winning mentality.
"Players come and go in football, that's the nature of the game. Some will remain at a club their whole career, like Stevie Gerrard, and some will move.
"It doesn't matter.
"We will still have the mentality to move forward, be an exciting squad, playing attacking football and that's the Liverpool way. We will continue to do that."
