The Moscow talks are seen as an attempt by the leaders of both countries to iron out differences.
Russian President Vladimir Putin initially backed Mr Yushchenko's rival Viktor Yanukovych - whose victory was marred by fraud and later annulled.
Mr Yushchenko on Sunday said Ukraine's future was in the EU. He will also tour Central and Western Europe this week.
Mr Yushchenko beat Mr Yanukovych in a re-run of November's disputed poll.
The 26 December repeat run-off was held after the Supreme Court ruled that the original vote - officially won by Mr Yanukovych - had been rigged.
Independent election observers said the re-run had been much fairer than earlier rounds.
'Honest talks'
By making his first foreign visit to Moscow, Mr Yushchenko keeps a promise given at the height of Ukraine's longest-ever presidential campaign.
"We need to talk without delay and honestly," he told reporters in Ukraine at the time.
His camp complained about what it perceived as meddling by Mr Putin, who twice visited Ukraine during the campaign to show support for Mr Yanukovych.
"I really disliked the fact that Russia campaigned in favour of one candidate," Mr Yushchenko told the Russian daily Izvestia last month.
Mr Putin also angered Mr Yushchenko's backers by congratulating Mr Yanukovych with the victory in the original run-off without waiting for the final results to be announced.
The Russian leader later retracted his congratulatory telegram, saying he was ready to work with whoever would be elected Ukraine's president.
Mr Yushchenko's Moscow visit is also seen as an attempt to placate Ukraine's predominantly Russian speaking south-eastern regions, where Mr Yanukovych enjoys strong backing, analysts say.
They say Mr Yushchenko wants to demonstrate his pragmatism in emphasising the importance of having good relations with Ukraine's biggest trade partner.
'Orange Kiev'
Mr Yushchenko's swearing-in on Sunday in the capital, Kiev, was watched by eight heads of state, along with US Secretary of State Colin Powell.
In contrast, Russia had relatively low-level representation - Sergei Mironov, the head of the upper house of parliament.
The date for the inauguration was set only this week after the Supreme Court dismissed a final appeal by Mr Yanukovych against the result.
The gathering in Independence Square allowed Mr Yushchenko to thank those who took part in what has been called the Orange Revolution - after the colour of Mr Yushchenko's camp.
The president told the massive crowd: "This is a victory of freedom over tyranny... We are not on the outskirts of Europe, we are at the centre of Europe."