The Reds have progressed each season under Jurgen Klopp since the German's arrival in 2015, but they are yet to win a trophy.
They can change that fact on June 1 when they take on Tottenham Hotspur in the Champions League final, and Sturridge says a victory will lead to more silverware for the Merseyside club.
“It (The squad) has improved a lot," he said. “But the main thing everybody wants is success and trophies.
“I feel we need to take the next step and push to the next level. Winning a trophy – which is the biggest hurdle – allows you to win a lot more because you know how to win.
“That's the thing we're lacking at the moment, jumping over that last hurdle to provide the first trophy under the manager."