Sturridge, recruited last week from Chelsea, took only seven minutes of his Reds debut to score on Sunday, in the FA Cup third-round tie at Mansfield.
The 23-year-old 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."