Sturridge, recruited last week from Chelsea, took only seven minutes of his Reds debut to score on Sunday, latching on to fellow England international Jonjo Shelvey's through-ball to net the opener in the FA Cup third-round tie at Mansfield.
The 23-year-old failed to convert further chances that followed, though, and Sturridge - who last played a senior match in November - hopes an improvement in his physical condition will help him demonstrate a more clinical edge.
He told the Liverpool Echo: "I haven't had much training.
"I've only had three sessions with the lads and then a light session on Saturday before the game.
"I am lacking sharpness in front of goal. I missed a few chances but hopefully once I get my fitness those ones that I missed will go in.
"It was a great pass from Jonjo - the vision and the weight of the pass for the goal was perfect. He made it quite easy for me to take the shot first time.
"We haven't had much time to work on stuff but we were both on the same wavelength. We get on well on and off the field.
"All the lads have been great with me and they've made it easy for me to settle in over the past few days.
"I want to say thank you to them for making me feel so welcome. I look forward to playing with them for many years to come."
Liverpool went on to beat Mansfield 2-1, with Luis Suarez netting their second goal on 59 minutes in controversial circumstances and Matt Green then pulling one back for the non-league hosts with 11 minutes of normal time remaining.
The ball struck Suarez's arm before crossing the line for his winner and regarding the incident, Sturridge, who had been substituted for the Uruguayan, said: "I didn't have a great view of our second goal from the bench and I haven't seen it again.
"I didn't see exactly what happened but the goal stood and we're through to the next round which is what really matters."