Viktor Zubkov, Russia's prime minister, resigned shortly after Medvedev's inauguration, clearing the way for Putin to be appointed in the post.
Putin, who was barred from running for a third consecutive presidential term by Russia's constitution, is expected to exercise considerable power as an adviser to Medvedev.
Rule of law
Jonah Hull, Al Jazeera's correspondent in Moscow, said Medvedev has "pretty strong ideas of what he wants to try and achieve for Russia".
"Perhaps the most striking note [of Medvedev's speech] was the emphasis that he put on human rights and freedoms," he said.
"That's a little in contrast to the tone in things we've seen under Vladimir Putin."
Medvedev said he will aim to create "new and wide opportunities of self-fulfilment for citizens - citizens who are free and responsible both for their own success and the flourishing of the entire country."
|Many Russians think that Putin will manage|
Medvedev's presidency [AFP]
"It is such people who create a really worthy nation and are the sources of a strong state, a state which today has the needed resources and a clear understanding of its national interests."
Highlighting his background as a lawyer, the new president said that he would work to ensure that Russian laws are applied fairly.
"I attach particular importance to the fundamental role of the law, on which both our state and our civil society are based," he said.
"We must achieve a sincere respect for the law and overcome legal nihilism."
Sergei Strokan, a Russian journalist, told Al Jazeera that he was particularly interested by Medvedev's emphasis on abiding by the rule of law.
"At the start of Putin's presidency, he also was speaking about law, but he said 'dictatorship of law'. There was no mention of dictatorship in Medvedev's speech," he said.
"But Medvedev is a lawyer himself and he is emphasising that law has to awaken Russia."
Medvedev thanked Putin for his leadership during his inauguration and said he would count on his support in the future.
Medvedev's inauguration caps a rise from being a Putin-era bureaucrat to the leader of the world's largest energy producing country.
He takes over from a leader who presided over strong economic growth in Russia.
The country has recovered from an economic collapse in the early 1990s to become the world's 10th largest economy, with a $1.3trn gross domestic product.
Economic growth has risen from 7 per cent in 2006 to 8.1 per cent in 2007, and the average monthly salary has risen from $80 to $640 over the course of Putin's two terms as president.
Medvedev campaigned in March's presidential election on a promise to follow a plan engineered by Putin.
Medvedev's first important act will be to formally name Putin as Russia's prime minister, after which they will both attend a military parade at Red Square on Friday.
Two thirds of Russians believe that Putin will control Medvedev despite being in a less prestigious office, according to a poll released in April by the Levada Centre, a public opinion survey organisation.