Mr Drakeford also confirmed that outdoor cinemas and mobile hairdressers would be able to reopen from next week, with most indoor visitor attractions also able to welcome back visitors.
Meanwhile, community centres, outdoor gyms and playgrounds are set to reopen on July 20, in time for the school summer holidays.
Mr Drakeford has also signalled to businesses in the beauty industry, including tattooists and nail salons, that they can begin to prepare for reopening on July 27, if conditions allow for reopening.
Meanwhile, pubs, cafes, restaurants and bars could reopen indoors on August 3.
The First Minister said: “Together we are making good progress to tackle the spread of this virus. It is thanks to the efforts we have made together that we are able to lift the restrictions and reopen more parts of our society and economy.
“However, the threat from coronavirus has not gone away and only by all of us acting responsibly will we be able to keep Wales safe. This means maintaining social distancing, thinking carefully about where we go and why.”
He added: “We have used our headroom to continue reopening our economy and society in Wales. We have also used part of our headroom to enable the Welsh NHS to provide more essential care and treatment for people – services, which were suspended in March to allow the health service to prepare for coronavirus.
“Our ability to carry on lifting the restrictions rests on everyone in Wales – we need everyone’s help to continue following the rule to keep levels of coronavirus at an absolute minimum.”
The next formal review of lockdown restrictions is due by July 30.
Welsh Government has said it’s in discussions with local authorities about how gyms, leisure centres, fitness studios and swimming pools can safely reopen.
Plaid Cymru leader Adam Price said: “The Welsh Government is exercising its devolved powers to, rightly, take a cautious approach to the easing of lockdown.”
He also called for a “Test, Trace and Protect programme that is able to keep up with increased demand”.
Welsh Conservative leader Paul Davies expressed disappointment at the “unnecessary delay” in implementing the measures.
Mr Davies said: “We see no reason why re-opening campsites and beauty salons, and re-starting house viewings can’t be brought forward”.
He added: “We are disappointed that there’s still no clarity on face coverings, relaxing social distancing, or driving tests and lessons in Wales.
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