She added that many Scots were angry that their Remain vote had been "assimilated" by the SNP and turned into a "proxy vote" for independence.
Labour was beaten into third place by the Conservatives in Scotland, where the SNP was comfortably the largest party despite failing to take control of target councils.
The SNP came top of the poll with 32.3% of first preference votes cast, with the Scottish Conservatives in second with 25.3%.
That said, the turnout last week was 46.9 per cent, some way short of what will be recorded in next month's general election. They chose to fight the election on the issue of an independence referendum, they talked about nothing else, they didn't have any policies for local government.
"And it was in the north-east where the Conservatives - whose tally was up by 12 points across Scotland as a whole, enough to put them five points ahead of Labour - registered their biggest advances". But having stood only 43 candidates in this election, they needed all of them to win.
"It's time to focus on this crucial general election campaign - and we can take absolutely nothing for granted", May will tell the event on Monday.
Meeting victorious SNP candidates in Glasgow, where Labour lost overall control of the council for the first time since 1980, Sturgeon told Sky News on Saturday: "Yes, by their standards, the Tories did well - and I take nothing away from them - but they did well entirely at the expense of the Labour party".
However, if voting patterns are similar at the General Election on June 8, the surge in Conservative support could see Ruth Davidson's party oust some SNP MPs from Westminster.
The full results of last week's council elections have been published by the Electoral Management Board for Scotland.
He said: "There was no outright victor, the SNP have one more member, and that means we have to have a totally different mind-set".
In a series of eyebrow-raising results, the Tories also won a seat in Cowdenbeath, where previous year the UK's last elected Communist stood down, and even in Ravenscraig.
And this, I suggest, is a problem for the first minister. And although the SNP will certainly "win" the general election in Scotland, just as they have the local government elections, they now have cause for concern.
"We are determined to speak up for people in Scotland who are beginning to find their voice". "I have to say I'm delighted, the effect is to galvanise the SNP support".
Ms Sturgeon said: "I'm over the moon that we are the largest party in Glasgow and are about to form an administration in the city".
"I will always put the people of Moray and the people of Scotland first". "We will not be a government that presides over food bank Britain or asks teachers to go on the school gates and collect funds to pay the teachers' wages", he said. Download it today and continue to enjoy STV News wherever you are.