Cook, 29, was absent as England lost their one-off Twenty20 international against the same opponents on Tuesday.
But the Essex batsman returns for the five-match one-day series, which starts at The Oval on Thursday.
“I think ‘getting going’ are the right words,” Cook said. “We just want to play some cricket. We want to start the journey of the next year or so.”
Since returning from a winter during which they were beaten 5-0 by Australia and lost to the Netherlands in the World Twenty20, Cook has been eager to begin working with new England coach Peter Moores.
Moores, 51, was reappointed after Andy Flower resigned following the Ashes whitewash.
His partnership with Cook began with a rain-affected win over Scotland in Aberdeen and continues with the Test and one-day series against Sri Lanka and India this summer.
Cook said: “Against Scotland, we put in a decent performance, but the summer starts now.”
He added England had played “really well” for much of the nine-run Twenty20 defeat by Sri Lanka on Tuesday, but admitted: “At the crucial moments, we didn’t quite get our skills right – whether it was with the ball, bat or fielding.
“Maybe it’s the habit of not winning many games recently. These are games which – when you’re flying, as Sri Lanka are as world champions – you find a way of winning.”
Cook acknowledged victory would have boosted morale in the England squad, but felt they still had cause for optimism.
“It does help when you’ve beaten a side, certainly in our camp, you’d feel a lot better,” he said.
“But when you’re on the end of a loss, with a few different players coming back in, you won’t look too much into the Twenty20 game.”
While England are choosing not to read too much into Tuesday’s result, Sri Lanka have gained renewed self confidence and belief from their triumph.
“To come and beat England in England was a great thing,” said Marvan Atapattu, Sri Lanka’s coach.
“We needed that win to tell ourselves we are good enough to compete against these guys on their home soil.”
He admitted England’s change of personnel both in the squad and backroom staff made them a team difficult to predict.
“From England’s point of view, they have gone through a couple of shaky series and have a new management and a couple of new players,” he added.
“They could be different in their approach, but we are most concerned with how we perform in English conditions to meet the challenge.”
Squads for the one-day series:
England squad for first three ODIs: Alastair Cook (capt), Ian Bell, Joe Root, Gary Ballance, Eoin Morgan, Jos Buttler (wkt), Ravi Bopara, Chris Jordan, James Tredwell, Harry Gurney, Jimmy Anderson, Tim Bresnan, Chris Woakes, Michael Carberry.
Sri Lanka squad for the five-match ODI series: Angelo Mathews (capt), Tillakaratne Dilshan, Kusal Perera, Kumar Sangakkara (wkt), Mahela Jayawardene, Lahiru Thirimanne, Thisara Perera, Nuwan Kulasekara, Suranga Lakmal, Lasith Malinga, Ajantha Mendis, Dinesh Chandimal, Ashan Priyanjan, Sachithra Senanayake, Chathuranga de Silva,