The Prime Minister said “now everyone must play their part”, as England pushes ahead with the third stage of the road map out of lockdown on Monday.
Pubs and restaurants will be able to welcome customers back indoors, household visits will resume and the foreign holiday ban will end.
Hugs and other physical contact between households will be permitted for the first time since restrictions began more than a year ago.
But the measures are being eased as top scientists warn of a “perilous moment”, with the Indian variant feared to be as much as 50% more transmissible than the Kent strain.
In a statement, Mr Johnson said: “Together we have reached another milestone in our roadmap out of lockdown, but we must take this next step with a heavy dose of caution.
“We are keeping the spread of the variant first identified in India under close observation and taking swift action where infection rates are rising.”
He said the “current data does not indicate unsustainable pressure on the NHS”, and that second vaccine doses are being accelerated to give the greatest protection to the most vulnerable.
He added: “But now everyone must play their part – by getting tested twice a week, coming forward for your vaccine when called and remembering hands, face, space and fresh air.
“I urge everyone to be cautious and take responsibility when enjoying new freedoms today in order to keep the virus at bay.”
Ministers are hoping surge testing and vaccines will allow a safe opening up of the nation, with jabs due to be extended to the over-35s this week.
But Health Secretary Matt Hancock did not rule out the possibility of imposing local lockdowns in areas such as Bolton to tackle the Indian variant, which he warned could “spread like wildfire”.
He said there are more than 1,300 cases of the Indian variant of concern, which is “relatively widespread in small numbers” and is becoming “the dominant strain” in Bolton and Blackburn.
But offering good news over plans to ease restrictions without unleashing a fresh wave of infections and deaths, Mr Hancock told Sky’s Sophy Ridge On Sunday there is “new very early data” from Oxford University giving confidence that existing vaccines work against the variant.
He added: “That means that we can stay on course with our strategy of using the vaccine to deal with the pandemic and opening up carefully and cautiously, but we do need to be really very vigilant to the spread of the disease.
“We have a high degree of confidence that the vaccine will overcome.”
Cinemas, hotels and B&Bs will also be allowed to reopen on Monday, as the “rule of six” will be applied to indoor meetings, while physical distancing ends between friends and family.
Sir John Bell, Oxford’s regius professor of medicine, said the result of lab experiments investigating whether the vaccine neutralises the Indian variant “looks okay”.
“It’s not perfect, but it’s not catastrophically bad,” he told Times Radio, adding there is only “a slight reduction in the ability to neutralise the virus”.
Mr Hancock urged people to come forward for their jabs when eligible as he said the “vast majority” of people hospitalised with the strain in Bolton have not taken the vaccine.
The Health Secretary said there had been no known deaths from the Indian variant in Bolton of somebody who has received both jabs.
Five people have been in hospital with it after receiving their first vaccine dose, while one person who had received both doses had been admitted “but that person was frail”, he told The Andrew Marr Show.
With surge testing also under way in areas of Blackburn, Sefton and London, Government scientific adviser Professor Sir Mark Walport warned it will be “extremely important” to keep an eye on the numbers over the next few weeks.
He told Ridge: “The problem is that the data takes some while to emerge, which is why there’s… every grounds to be very cautious about the ability to open up in June.
“But I mean I think it’s fair to say it is a perilous moment. We have a variant that shows good evidence of being more transmissible and possibly significantly more transmissible.”
Chair of the Commons Home Affairs Committee Yvette Cooper suggested the foreign holiday ban should not end as scheduled on Monday, as it is replaced by a traffic light system of restrictions.
The Labour MP told the BBC’s Marr: “The Government needs to slow down its plans. I don’t understand why it’s lifting some of its international travel restrictions tomorrow. I think they should be being much more cautious about that.”
Though the Government is continuing with Monday’s relaxation, Mr Johnson has warned the Indian variant could jeopardise plans to end legal restrictions on June 21.
Clubs and gatherings of more than 30 people indoors will remain prohibited until the fourth and final phase of the road map.