The leader of the opposition pledged to engage “constructively” with the Conservatives, but criticised ministers for being too slow at explaining why the UK is “so far behind” on testing.
He also called for “blockages in the system” which are delaying the delivery of protective equipment – known as PPE – to front-line workers to be “addressed quickly”.
Writing in the Sunday Times after his landslide victory in the contest to replace Jeremy Corbyn, Sir Keir said: “There will be many times when, and there are many issues upon which, I will fundamentally disagree with the Prime Minister.
“However, there will also be times when Labour can – and must – engage constructively with the Government.
“Now is such a time. Coronavirus is a national emergency. It is also a global emergency. Everyone is anxious about what the next few months will bring, but we know we must be resolute in our determination to see this virus defeated, as it will be.
“I want to see the Government succeed in this: to save lives and protect livelihoods. This is a national effort and all of us should be asking what more we can do.”
He said Labour would “do our bit to offer solutions” but would also “speak for those who have been ignored”, and expose mistakes where they are found “to ensure that they are rectified as soon as possible”.
Sir Keir wrote: “And let’s be honest, serious mistakes have been made.
“The public is placing an enormous trust in the Government at the moment: it is vital that that trust is met with openness and transparency about those mistakes and the decisions that have been made.”
As well as his calls on testing and PPE, Sir Keir said the UK should build vaccination centres in towns and cities across the country so “the minute a vaccine becomes available, we can begin to protect the entire population”.
He also said ministers should publish their exit strategy from the measures to defeat coronavirus.
“We should know what that exit strategy is, when the restrictions might be lifted and what the plan is for economic recovery to protect those who have been hardest hit,” he said.
“There will be many more difficult days ahead. Great sacrifices must be made because of a crisis that was unimaginable only a few months ago. But Britain is a great country and we will get through this.”
Coronavirus: Everything you need to know