He takes over from outgoing president KR Narayanan after being elected to the post last week.
In his acceptance speech Dr Kalam called for India to be turned into a developed nation by focusing on education.
The retired architect of India's missile programme, Dr Kalam was backed for the post by most of the major political parties, including the governing coalition and the opposition Congress party.
Dr Kalam said India faced many challenges, including cross-border terrorism, internal conflicts and unemployment.
These needed "a vision to ensure focused action of one billion citizens of this great country with varied capabilities".
"India should become a developed country which is free from poverty, illiteracy and unemployment and is buoyant with economic prosperity, national security and internal harmony," he added.
Colourful ceremony
Dr Kalam took the oath of office in a colourful ceremony in the Indian parliament, followed by a 21-gun salute.
He was escorted to the parliament building by President Narayanan, flanked by the presidential bodyguard mounted on horses.
Inside the high-domed central hall, he was sworn in by the Chief Justice in a formal ceremony attended by Prime Minister Atal Behari Vajpayee, Opposition leader Sonia Gandhi and some 700 MPs.
At Mr Kalam's special request, 100 school children were invited to attend the inauguration.
On Wednesday, in a farewell speech President Narayanan made a strong appeal for India to hold on to its tradition of religious tolerance.
"It is important for us today to introspect and realise that what makes India's unity and democracy credible and enduring is this precious tradition of tolerance," he said in a nationally televised address.
Mr Narayanan's comments come after religious violence in the western state of Gujarat earlier this year claimed nearly a thousand lives, mostly Muslims.
The new head of state was elected by an electoral college of about 5,000 central and state legislators.
The president of India is a titular head of state with few actual powers but his authority to decide which party or individual should be asked to form the central government after general elections gives his post significance.
This is especially the case when no single party has an overall majority in parliament.
Unusual choice
Dr Kalam is acknowledged as the driving force behind India's quest for cutting-edge defence technologies, and has helped turn India into a nuclear power.
He has no political experience and was seen by some as an odd choice for the presidency.
But others feel he will bring refreshing colour to the largely ceremonial post.
With his signature long, unruly silver hair and casual clothes, Dr Kalam has spent most of the past year teaching and speaking passionately about integrity and values.
He is known for his strong personal discipline, and is a strict vegetarian, teetotaller and a bachelor.
Although a Muslim, he is well versed in Hindu scriptures.